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
|
syl5ibr |
⊢ ( ( 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
|
syl6ibr |
⊢ ( ( 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 ) ) |