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Metamath Proof Explorer

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