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Metamath Proof Explorer

Theorem fidomtri

Description: Trichotomy of dominance without AC when one set is finite. (Contributed by Stefan O'Rear, 30-Oct-2014) (Revised by Mario Carneiro, 27-Apr-2015)

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

Proof

Step Hyp Ref Expression
1 domnsym ⊢ ( 𝐴 ≼ 𝐵 → ¬ 𝐵 ≺ 𝐴 )
2 finnum ⊢ ( 𝐴 ∈ Fin → 𝐴 ∈ dom card )
3 2 adantr ⊢ ( ( 𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉 ) → 𝐴 ∈ dom card )
4 finnum ⊢ ( 𝐵 ∈ Fin → 𝐵 ∈ dom card )
5 domtri2 ⊢ ( ( 𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ) → ( 𝐴 ≼ 𝐵 ↔ ¬ 𝐵 ≺ 𝐴 ) )
6 3 4 5 syl2an ⊢ ( ( ( 𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉 ) ∧ 𝐵 ∈ Fin ) → ( 𝐴 ≼ 𝐵 ↔ ¬ 𝐵 ≺ 𝐴 ) )
7 6 biimprd ⊢ ( ( ( 𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉 ) ∧ 𝐵 ∈ Fin ) → ( ¬ 𝐵 ≺ 𝐴 → 𝐴 ≼ 𝐵 ) )
8 isinffi ⊢ ( ( ¬ 𝐵 ∈ Fin ∧ 𝐴 ∈ Fin ) → ∃ 𝑎 𝑎 : 𝐴 –1-1→ 𝐵 )
9 8 ancoms ⊢ ( ( 𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin ) → ∃ 𝑎 𝑎 : 𝐴 –1-1→ 𝐵 )
10 9 adantlr ⊢ ( ( ( 𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉 ) ∧ ¬ 𝐵 ∈ Fin ) → ∃ 𝑎 𝑎 : 𝐴 –1-1→ 𝐵 )
11 brdomg ⊢ ( 𝐵 ∈ 𝑉 → ( 𝐴 ≼ 𝐵 ↔ ∃ 𝑎 𝑎 : 𝐴 –1-1→ 𝐵 ) )
12 11 ad2antlr ⊢ ( ( ( 𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉 ) ∧ ¬ 𝐵 ∈ Fin ) → ( 𝐴 ≼ 𝐵 ↔ ∃ 𝑎 𝑎 : 𝐴 –1-1→ 𝐵 ) )
13 10 12 mpbird ⊢ ( ( ( 𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉 ) ∧ ¬ 𝐵 ∈ Fin ) → 𝐴 ≼ 𝐵 )
14 13 a1d ⊢ ( ( ( 𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉 ) ∧ ¬ 𝐵 ∈ Fin ) → ( ¬ 𝐵 ≺ 𝐴 → 𝐴 ≼ 𝐵 ) )
15 7 14 pm2.61dan ⊢ ( ( 𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉 ) → ( ¬ 𝐵 ≺ 𝐴 → 𝐴 ≼ 𝐵 ) )
16 1 15 impbid2 ⊢ ( ( 𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉 ) → ( 𝐴 ≼ 𝐵 ↔ ¬ 𝐵 ≺ 𝐴 ) )