Step |
Hyp |
Ref |
Expression |
1 |
|
findcard2s.1 |
⊢ ( 𝑥 = ∅ → ( 𝜑 ↔ 𝜓 ) ) |
2 |
|
findcard2s.2 |
⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜒 ) ) |
3 |
|
findcard2s.3 |
⊢ ( 𝑥 = ( 𝑦 ∪ { 𝑧 } ) → ( 𝜑 ↔ 𝜃 ) ) |
4 |
|
findcard2s.4 |
⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜏 ) ) |
5 |
|
findcard2s.5 |
⊢ 𝜓 |
6 |
|
findcard2s.6 |
⊢ ( ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) → ( 𝜒 → 𝜃 ) ) |
7 |
6
|
ex |
⊢ ( 𝑦 ∈ Fin → ( ¬ 𝑧 ∈ 𝑦 → ( 𝜒 → 𝜃 ) ) ) |
8 |
|
snssi |
⊢ ( 𝑧 ∈ 𝑦 → { 𝑧 } ⊆ 𝑦 ) |
9 |
|
ssequn1 |
⊢ ( { 𝑧 } ⊆ 𝑦 ↔ ( { 𝑧 } ∪ 𝑦 ) = 𝑦 ) |
10 |
8 9
|
sylib |
⊢ ( 𝑧 ∈ 𝑦 → ( { 𝑧 } ∪ 𝑦 ) = 𝑦 ) |
11 |
|
uncom |
⊢ ( { 𝑧 } ∪ 𝑦 ) = ( 𝑦 ∪ { 𝑧 } ) |
12 |
10 11
|
eqtr3di |
⊢ ( 𝑧 ∈ 𝑦 → 𝑦 = ( 𝑦 ∪ { 𝑧 } ) ) |
13 |
|
vex |
⊢ 𝑦 ∈ V |
14 |
13
|
eqvinc |
⊢ ( 𝑦 = ( 𝑦 ∪ { 𝑧 } ) ↔ ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝑥 = ( 𝑦 ∪ { 𝑧 } ) ) ) |
15 |
12 14
|
sylib |
⊢ ( 𝑧 ∈ 𝑦 → ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝑥 = ( 𝑦 ∪ { 𝑧 } ) ) ) |
16 |
2
|
bicomd |
⊢ ( 𝑥 = 𝑦 → ( 𝜒 ↔ 𝜑 ) ) |
17 |
16 3
|
sylan9bb |
⊢ ( ( 𝑥 = 𝑦 ∧ 𝑥 = ( 𝑦 ∪ { 𝑧 } ) ) → ( 𝜒 ↔ 𝜃 ) ) |
18 |
17
|
exlimiv |
⊢ ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝑥 = ( 𝑦 ∪ { 𝑧 } ) ) → ( 𝜒 ↔ 𝜃 ) ) |
19 |
15 18
|
syl |
⊢ ( 𝑧 ∈ 𝑦 → ( 𝜒 ↔ 𝜃 ) ) |
20 |
19
|
biimpd |
⊢ ( 𝑧 ∈ 𝑦 → ( 𝜒 → 𝜃 ) ) |
21 |
7 20
|
pm2.61d2 |
⊢ ( 𝑦 ∈ Fin → ( 𝜒 → 𝜃 ) ) |
22 |
1 2 3 4 5 21
|
findcard2 |
⊢ ( 𝐴 ∈ Fin → 𝜏 ) |