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	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐴 ≼ 𝐵 ↔ ¬ 𝐵 ≺ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		numth3	⊢ ( 𝐴 ∈ 𝑉 → 𝐴 ∈ dom card )
2		numth3	⊢ ( 𝐵 ∈ 𝑊 → 𝐵 ∈ dom card )
3		domtri2	⊢ ( ( 𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ) → ( 𝐴 ≼ 𝐵 ↔ ¬ 𝐵 ≺ 𝐴 ) )
4	1 2 3	syl2an	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐴 ≼ 𝐵 ↔ ¬ 𝐵 ≺ 𝐴 ) )