Theorem domtri

Description: Trichotomy law for dominance and strict dominance. This theorem is equivalent to the Axiom of Choice. (Contributed by NM, 4-Jan-2004) (Revised by Mario Carneiro, 30-Apr-2015)

	Ref	Expression
Assertion	domtri	$$⊢\left(A\in V\wedge B\in W\right)\to \left(A\preccurlyeq B\leftrightarrow \neg B\prec A\right)$$

Proof

Step	Hyp	Ref	Expression
1		numth3	$$⊢A\in V\to A\in \mathrm{dom}\mathrm{card}$$
2		numth3	$$⊢B\in W\to B\in \mathrm{dom}\mathrm{card}$$
3		domtri2	$$⊢\left(A\in \mathrm{dom}\mathrm{card}\wedge B\in \mathrm{dom}\mathrm{card}\right)\to \left(A\preccurlyeq B\leftrightarrow \neg B\prec A\right)$$
4	1 2 3	syl2an	$$⊢\left(A\in V\wedge B\in W\right)\to \left(A\preccurlyeq B\leftrightarrow \neg B\prec A\right)$$