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Metamath Proof Explorer

Theorem fidomtri2

Description: Trichotomy of dominance without AC when one set is finite. (Contributed by Stefan O'Rear, 30-Oct-2014) (Revised by Mario Carneiro, 7-May-2015)

Ref Expression
Assertion fidomtri2 ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ Fin ) → ( 𝐴 ≼ 𝐵 ↔ ¬ 𝐵 ≺ 𝐴 ) )

Proof

Step Hyp Ref Expression
1 domnsym ⊢ ( 𝐴 ≼ 𝐵 → ¬ 𝐵 ≺ 𝐴 )
2 sdomdom ⊢ ( 𝐴 ≺ 𝐵 → 𝐴 ≼ 𝐵 )
3 2 con3i ⊢ ( ¬ 𝐴 ≼ 𝐵 → ¬ 𝐴 ≺ 𝐵 )
4 fidomtri ⊢ ( ( 𝐵 ∈ Fin ∧ 𝐴 ∈ 𝑉 ) → ( 𝐵 ≼ 𝐴 ↔ ¬ 𝐴 ≺ 𝐵 ) )
5 4 ancoms ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ Fin ) → ( 𝐵 ≼ 𝐴 ↔ ¬ 𝐴 ≺ 𝐵 ) )
6 3 5 imbitrrid ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ Fin ) → ( ¬ 𝐴 ≼ 𝐵 → 𝐵 ≼ 𝐴 ) )
7 ensym ⊢ ( 𝐵 ≈ 𝐴 → 𝐴 ≈ 𝐵 )
8 endom ⊢ ( 𝐴 ≈ 𝐵 → 𝐴 ≼ 𝐵 )
9 7 8 syl ⊢ ( 𝐵 ≈ 𝐴 → 𝐴 ≼ 𝐵 )
10 9 con3i ⊢ ( ¬ 𝐴 ≼ 𝐵 → ¬ 𝐵 ≈ 𝐴 )
11 6 10 jca2 ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ Fin ) → ( ¬ 𝐴 ≼ 𝐵 → ( 𝐵 ≼ 𝐴 ∧ ¬ 𝐵 ≈ 𝐴 ) ) )
12 brsdom ⊢ ( 𝐵 ≺ 𝐴 ↔ ( 𝐵 ≼ 𝐴 ∧ ¬ 𝐵 ≈ 𝐴 ) )
13 11 12 imbitrrdi ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ Fin ) → ( ¬ 𝐴 ≼ 𝐵 → 𝐵 ≺ 𝐴 ) )
14 13 con1d ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ Fin ) → ( ¬ 𝐵 ≺ 𝐴 → 𝐴 ≼ 𝐵 ) )
15 1 14 impbid2 ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ Fin ) → ( 𝐴 ≼ 𝐵 ↔ ¬ 𝐵 ≺ 𝐴 ) )