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Metamath Proof Explorer

Theorem entric

Description: Trichotomy of equinumerosity and strict dominance. This theorem is equivalent to the Axiom of Choice. Theorem 8 of Suppes p. 242. (Contributed by NM, 4-Jan-2004)

Ref Expression
Assertion entric ⊢ A ∈ V ∧ B ∈ W → A ≺ B ∨ A ≈ B ∨ B ≺ A

Proof

Step Hyp Ref Expression
1 domtri ⊢ A ∈ V ∧ B ∈ W → A ≼ B ↔ ¬ B ≺ A
2 1 biimprd ⊢ A ∈ V ∧ B ∈ W → ¬ B ≺ A → A ≼ B
3 brdom2 ⊢ A ≼ B ↔ A ≺ B ∨ A ≈ B
4 2 3 imbitrdi ⊢ A ∈ V ∧ B ∈ W → ¬ B ≺ A → A ≺ B ∨ A ≈ B
5 4 con1d ⊢ A ∈ V ∧ B ∈ W → ¬ A ≺ B ∨ A ≈ B → B ≺ A
6 5 orrd ⊢ A ∈ V ∧ B ∈ W → A ≺ B ∨ A ≈ B ∨ B ≺ A
7 df-3or ⊢ A ≺ B ∨ A ≈ B ∨ B ≺ A ↔ A ≺ B ∨ A ≈ B ∨ B ≺ A
8 6 7 sylibr ⊢ A ∈ V ∧ B ∈ W → A ≺ B ∨ A ≈ B ∨ B ≺ A