Step |
Hyp |
Ref |
Expression |
1 |
|
prprc1 |
⊢ ( ¬ 𝐴 ∈ V → { 𝐴 , 𝐵 } = { 𝐵 } ) |
2 |
|
snfi |
⊢ { 𝐵 } ∈ Fin |
3 |
1 2
|
eqeltrdi |
⊢ ( ¬ 𝐴 ∈ V → { 𝐴 , 𝐵 } ∈ Fin ) |
4 |
|
prprc2 |
⊢ ( ¬ 𝐵 ∈ V → { 𝐴 , 𝐵 } = { 𝐴 } ) |
5 |
|
snfi |
⊢ { 𝐴 } ∈ Fin |
6 |
4 5
|
eqeltrdi |
⊢ ( ¬ 𝐵 ∈ V → { 𝐴 , 𝐵 } ∈ Fin ) |
7 |
|
2onn |
⊢ 2o ∈ ω |
8 |
|
simp1 |
⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ∧ ¬ 𝐴 = 𝐵 ) → 𝐴 ∈ V ) |
9 |
|
simp2 |
⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ∧ ¬ 𝐴 = 𝐵 ) → 𝐵 ∈ V ) |
10 |
|
simp3 |
⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ∧ ¬ 𝐴 = 𝐵 ) → ¬ 𝐴 = 𝐵 ) |
11 |
8 9 10
|
enpr2d |
⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ∧ ¬ 𝐴 = 𝐵 ) → { 𝐴 , 𝐵 } ≈ 2o ) |
12 |
|
breq2 |
⊢ ( 𝑥 = 2o → ( { 𝐴 , 𝐵 } ≈ 𝑥 ↔ { 𝐴 , 𝐵 } ≈ 2o ) ) |
13 |
12
|
rspcev |
⊢ ( ( 2o ∈ ω ∧ { 𝐴 , 𝐵 } ≈ 2o ) → ∃ 𝑥 ∈ ω { 𝐴 , 𝐵 } ≈ 𝑥 ) |
14 |
7 11 13
|
sylancr |
⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ∧ ¬ 𝐴 = 𝐵 ) → ∃ 𝑥 ∈ ω { 𝐴 , 𝐵 } ≈ 𝑥 ) |
15 |
|
isfi |
⊢ ( { 𝐴 , 𝐵 } ∈ Fin ↔ ∃ 𝑥 ∈ ω { 𝐴 , 𝐵 } ≈ 𝑥 ) |
16 |
14 15
|
sylibr |
⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ∧ ¬ 𝐴 = 𝐵 ) → { 𝐴 , 𝐵 } ∈ Fin ) |
17 |
16
|
3expia |
⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( ¬ 𝐴 = 𝐵 → { 𝐴 , 𝐵 } ∈ Fin ) ) |
18 |
|
dfsn2 |
⊢ { 𝐴 } = { 𝐴 , 𝐴 } |
19 |
|
preq2 |
⊢ ( 𝐴 = 𝐵 → { 𝐴 , 𝐴 } = { 𝐴 , 𝐵 } ) |
20 |
18 19
|
eqtr2id |
⊢ ( 𝐴 = 𝐵 → { 𝐴 , 𝐵 } = { 𝐴 } ) |
21 |
20 5
|
eqeltrdi |
⊢ ( 𝐴 = 𝐵 → { 𝐴 , 𝐵 } ∈ Fin ) |
22 |
17 21
|
pm2.61d2 |
⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → { 𝐴 , 𝐵 } ∈ Fin ) |
23 |
3 6 22
|
ecase |
⊢ { 𝐴 , 𝐵 } ∈ Fin |