According to the Longman Dictionary of English Language and Culture which I own, integrity is strength and firmness of character or principle; honesty, trustworthiness and also a state of being whole and undivided; completeness.
Integrity basically means not compromising your principles, not engaging in unholy alliances and practising what you preach. It is a quality of character solely missed nowadays, and unfortunately, not only by unbelievers.
The Bible warns Christians not be unequally yoked because there can be nothing in common between light and darkness. There can be no compromise between a Christian and a hedonistic, nihilistic post-modern world view and there can also be nothing in common between traditional Western Christianity and modern progressive ideology.
Progressives always speak in terms of oppressors and oppressed and advocate for equality, which for them never means equal dignity but rather the redistribution of power and material goods.
As applied to the relationship between the sexes, progressives seeks equality between men and women which for them basically means that both sexes must have the same rights and responsibilities. This system doesn't work and it won't work, no matter how the social engineers are trying to push it, simply because men and women are different.
While feminists undoubtedly started it, there is no use for certain men to lament that women don't want to support themselves and prefer cushy government jobs to labouring in some ardous field, or that they often miss the same dedication men will demonstrate in their profession. You also can't say you are an anti-feminist and then expect your wife to have a career and support herself.
Patriarchy doesn't really mean the man staying at home drinking beer while his wife (or even wives as some of those propagating the new doctrine seem to support polygamy) is out there slaying dragons and bringing home the bacon.
In all of the world history, starting with Ancient Rome and Greece, extra rights meant also additional responsibilities. For instance, voting rights were tied up to the army service, that's why female suffrage is utterly dishonest since women aren't required to register for draft or fight in wars. However, the solution to the problem is not pushing women to become soldiers, as both feminists and certain men's rights activists want, but the restriction of the female suffrage as it should be obvious to anyone with half a brain cell, that the majority of women won't make good soldiers.
In the same manner, when the law made the man the head of the family, he also had to financially support his wife (I've written about this on my blog before). In the times of the Vikings, the government even had established the minimum bride price the man had to pay if he wished to marry, the reasoning behind it being that if a man was too poor to pay the minimum amount of money required by law he obviously wouldn't be able to support a family and hence had no business to marry.
Nowadays, both men and women want to have all sorts of rights, but nobody seems to want to have any duties. I'm not sure how long we as society can go on in the same manner, but something tells me that not very long any more. As Christians, we are called to be better than this and to live according to God's Law which clearly states that there are distinctive roles for men and women within the family and society.
Morally speaking, both men and women are fallen human beings who can be redeemed by God's grace and neither sex is better than the other. Men and women have equal dignity in Christ, but separate duties and responsibilities. A man should be respected for being the leader, protector and provider, and the woman for being the wife, mother and homemaker. Both roles are important and essential for the good functioning of society. The sex differences are good, because God created them.
Let's try to live our life according to a higher standard and show a good example of what being complementarian really means.