Step |
Hyp |
Ref |
Expression |
1 |
|
findcard2.1 |
⊢ ( 𝑥 = ∅ → ( 𝜑 ↔ 𝜓 ) ) |
2 |
|
findcard2.2 |
⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜒 ) ) |
3 |
|
findcard2.3 |
⊢ ( 𝑥 = ( 𝑦 ∪ { 𝑧 } ) → ( 𝜑 ↔ 𝜃 ) ) |
4 |
|
findcard2.4 |
⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜏 ) ) |
5 |
|
findcard2.5 |
⊢ 𝜓 |
6 |
|
findcard2.6 |
⊢ ( 𝑦 ∈ Fin → ( 𝜒 → 𝜃 ) ) |
7 |
|
isfi |
⊢ ( 𝑥 ∈ Fin ↔ ∃ 𝑤 ∈ ω 𝑥 ≈ 𝑤 ) |
8 |
|
breq2 |
⊢ ( 𝑤 = ∅ → ( 𝑥 ≈ 𝑤 ↔ 𝑥 ≈ ∅ ) ) |
9 |
8
|
imbi1d |
⊢ ( 𝑤 = ∅ → ( ( 𝑥 ≈ 𝑤 → 𝜑 ) ↔ ( 𝑥 ≈ ∅ → 𝜑 ) ) ) |
10 |
9
|
albidv |
⊢ ( 𝑤 = ∅ → ( ∀ 𝑥 ( 𝑥 ≈ 𝑤 → 𝜑 ) ↔ ∀ 𝑥 ( 𝑥 ≈ ∅ → 𝜑 ) ) ) |
11 |
|
breq2 |
⊢ ( 𝑤 = 𝑣 → ( 𝑥 ≈ 𝑤 ↔ 𝑥 ≈ 𝑣 ) ) |
12 |
11
|
imbi1d |
⊢ ( 𝑤 = 𝑣 → ( ( 𝑥 ≈ 𝑤 → 𝜑 ) ↔ ( 𝑥 ≈ 𝑣 → 𝜑 ) ) ) |
13 |
12
|
albidv |
⊢ ( 𝑤 = 𝑣 → ( ∀ 𝑥 ( 𝑥 ≈ 𝑤 → 𝜑 ) ↔ ∀ 𝑥 ( 𝑥 ≈ 𝑣 → 𝜑 ) ) ) |
14 |
|
breq2 |
⊢ ( 𝑤 = suc 𝑣 → ( 𝑥 ≈ 𝑤 ↔ 𝑥 ≈ suc 𝑣 ) ) |
15 |
14
|
imbi1d |
⊢ ( 𝑤 = suc 𝑣 → ( ( 𝑥 ≈ 𝑤 → 𝜑 ) ↔ ( 𝑥 ≈ suc 𝑣 → 𝜑 ) ) ) |
16 |
15
|
albidv |
⊢ ( 𝑤 = suc 𝑣 → ( ∀ 𝑥 ( 𝑥 ≈ 𝑤 → 𝜑 ) ↔ ∀ 𝑥 ( 𝑥 ≈ suc 𝑣 → 𝜑 ) ) ) |
17 |
|
en0 |
⊢ ( 𝑥 ≈ ∅ ↔ 𝑥 = ∅ ) |
18 |
5 1
|
mpbiri |
⊢ ( 𝑥 = ∅ → 𝜑 ) |
19 |
17 18
|
sylbi |
⊢ ( 𝑥 ≈ ∅ → 𝜑 ) |
20 |
19
|
ax-gen |
⊢ ∀ 𝑥 ( 𝑥 ≈ ∅ → 𝜑 ) |
21 |
|
nnon |
⊢ ( 𝑣 ∈ ω → 𝑣 ∈ On ) |
22 |
|
rexdif1en |
⊢ ( ( 𝑣 ∈ On ∧ 𝑤 ≈ suc 𝑣 ) → ∃ 𝑧 ∈ 𝑤 ( 𝑤 ∖ { 𝑧 } ) ≈ 𝑣 ) |
23 |
21 22
|
sylan |
⊢ ( ( 𝑣 ∈ ω ∧ 𝑤 ≈ suc 𝑣 ) → ∃ 𝑧 ∈ 𝑤 ( 𝑤 ∖ { 𝑧 } ) ≈ 𝑣 ) |
24 |
|
snssi |
⊢ ( 𝑧 ∈ 𝑤 → { 𝑧 } ⊆ 𝑤 ) |
25 |
|
uncom |
⊢ ( ( 𝑤 ∖ { 𝑧 } ) ∪ { 𝑧 } ) = ( { 𝑧 } ∪ ( 𝑤 ∖ { 𝑧 } ) ) |
26 |
|
undif |
⊢ ( { 𝑧 } ⊆ 𝑤 ↔ ( { 𝑧 } ∪ ( 𝑤 ∖ { 𝑧 } ) ) = 𝑤 ) |
27 |
26
|
biimpi |
⊢ ( { 𝑧 } ⊆ 𝑤 → ( { 𝑧 } ∪ ( 𝑤 ∖ { 𝑧 } ) ) = 𝑤 ) |
28 |
25 27
|
eqtrid |
⊢ ( { 𝑧 } ⊆ 𝑤 → ( ( 𝑤 ∖ { 𝑧 } ) ∪ { 𝑧 } ) = 𝑤 ) |
29 |
|
vex |
⊢ 𝑤 ∈ V |
30 |
29
|
difexi |
⊢ ( 𝑤 ∖ { 𝑧 } ) ∈ V |
31 |
|
breq1 |
⊢ ( 𝑦 = ( 𝑤 ∖ { 𝑧 } ) → ( 𝑦 ≈ 𝑣 ↔ ( 𝑤 ∖ { 𝑧 } ) ≈ 𝑣 ) ) |
32 |
31
|
anbi2d |
⊢ ( 𝑦 = ( 𝑤 ∖ { 𝑧 } ) → ( ( 𝑣 ∈ ω ∧ 𝑦 ≈ 𝑣 ) ↔ ( 𝑣 ∈ ω ∧ ( 𝑤 ∖ { 𝑧 } ) ≈ 𝑣 ) ) ) |
33 |
|
uneq1 |
⊢ ( 𝑦 = ( 𝑤 ∖ { 𝑧 } ) → ( 𝑦 ∪ { 𝑧 } ) = ( ( 𝑤 ∖ { 𝑧 } ) ∪ { 𝑧 } ) ) |
34 |
33
|
sbceq1d |
⊢ ( 𝑦 = ( 𝑤 ∖ { 𝑧 } ) → ( [ ( 𝑦 ∪ { 𝑧 } ) / 𝑥 ] 𝜑 ↔ [ ( ( 𝑤 ∖ { 𝑧 } ) ∪ { 𝑧 } ) / 𝑥 ] 𝜑 ) ) |
35 |
34
|
imbi2d |
⊢ ( 𝑦 = ( 𝑤 ∖ { 𝑧 } ) → ( ( ∀ 𝑥 ( 𝑥 ≈ 𝑣 → 𝜑 ) → [ ( 𝑦 ∪ { 𝑧 } ) / 𝑥 ] 𝜑 ) ↔ ( ∀ 𝑥 ( 𝑥 ≈ 𝑣 → 𝜑 ) → [ ( ( 𝑤 ∖ { 𝑧 } ) ∪ { 𝑧 } ) / 𝑥 ] 𝜑 ) ) ) |
36 |
32 35
|
imbi12d |
⊢ ( 𝑦 = ( 𝑤 ∖ { 𝑧 } ) → ( ( ( 𝑣 ∈ ω ∧ 𝑦 ≈ 𝑣 ) → ( ∀ 𝑥 ( 𝑥 ≈ 𝑣 → 𝜑 ) → [ ( 𝑦 ∪ { 𝑧 } ) / 𝑥 ] 𝜑 ) ) ↔ ( ( 𝑣 ∈ ω ∧ ( 𝑤 ∖ { 𝑧 } ) ≈ 𝑣 ) → ( ∀ 𝑥 ( 𝑥 ≈ 𝑣 → 𝜑 ) → [ ( ( 𝑤 ∖ { 𝑧 } ) ∪ { 𝑧 } ) / 𝑥 ] 𝜑 ) ) ) ) |
37 |
|
breq1 |
⊢ ( 𝑥 = 𝑦 → ( 𝑥 ≈ 𝑣 ↔ 𝑦 ≈ 𝑣 ) ) |
38 |
37 2
|
imbi12d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝑥 ≈ 𝑣 → 𝜑 ) ↔ ( 𝑦 ≈ 𝑣 → 𝜒 ) ) ) |
39 |
38
|
spvv |
⊢ ( ∀ 𝑥 ( 𝑥 ≈ 𝑣 → 𝜑 ) → ( 𝑦 ≈ 𝑣 → 𝜒 ) ) |
40 |
|
rspe |
⊢ ( ( 𝑣 ∈ ω ∧ 𝑦 ≈ 𝑣 ) → ∃ 𝑣 ∈ ω 𝑦 ≈ 𝑣 ) |
41 |
|
isfi |
⊢ ( 𝑦 ∈ Fin ↔ ∃ 𝑣 ∈ ω 𝑦 ≈ 𝑣 ) |
42 |
40 41
|
sylibr |
⊢ ( ( 𝑣 ∈ ω ∧ 𝑦 ≈ 𝑣 ) → 𝑦 ∈ Fin ) |
43 |
|
pm2.27 |
⊢ ( 𝑦 ≈ 𝑣 → ( ( 𝑦 ≈ 𝑣 → 𝜒 ) → 𝜒 ) ) |
44 |
43
|
adantl |
⊢ ( ( 𝑣 ∈ ω ∧ 𝑦 ≈ 𝑣 ) → ( ( 𝑦 ≈ 𝑣 → 𝜒 ) → 𝜒 ) ) |
45 |
42 44 6
|
sylsyld |
⊢ ( ( 𝑣 ∈ ω ∧ 𝑦 ≈ 𝑣 ) → ( ( 𝑦 ≈ 𝑣 → 𝜒 ) → 𝜃 ) ) |
46 |
39 45
|
syl5 |
⊢ ( ( 𝑣 ∈ ω ∧ 𝑦 ≈ 𝑣 ) → ( ∀ 𝑥 ( 𝑥 ≈ 𝑣 → 𝜑 ) → 𝜃 ) ) |
47 |
|
vex |
⊢ 𝑦 ∈ V |
48 |
|
vsnex |
⊢ { 𝑧 } ∈ V |
49 |
47 48
|
unex |
⊢ ( 𝑦 ∪ { 𝑧 } ) ∈ V |
50 |
49 3
|
sbcie |
⊢ ( [ ( 𝑦 ∪ { 𝑧 } ) / 𝑥 ] 𝜑 ↔ 𝜃 ) |
51 |
46 50
|
imbitrrdi |
⊢ ( ( 𝑣 ∈ ω ∧ 𝑦 ≈ 𝑣 ) → ( ∀ 𝑥 ( 𝑥 ≈ 𝑣 → 𝜑 ) → [ ( 𝑦 ∪ { 𝑧 } ) / 𝑥 ] 𝜑 ) ) |
52 |
30 36 51
|
vtocl |
⊢ ( ( 𝑣 ∈ ω ∧ ( 𝑤 ∖ { 𝑧 } ) ≈ 𝑣 ) → ( ∀ 𝑥 ( 𝑥 ≈ 𝑣 → 𝜑 ) → [ ( ( 𝑤 ∖ { 𝑧 } ) ∪ { 𝑧 } ) / 𝑥 ] 𝜑 ) ) |
53 |
|
dfsbcq |
⊢ ( ( ( 𝑤 ∖ { 𝑧 } ) ∪ { 𝑧 } ) = 𝑤 → ( [ ( ( 𝑤 ∖ { 𝑧 } ) ∪ { 𝑧 } ) / 𝑥 ] 𝜑 ↔ [ 𝑤 / 𝑥 ] 𝜑 ) ) |
54 |
53
|
imbi2d |
⊢ ( ( ( 𝑤 ∖ { 𝑧 } ) ∪ { 𝑧 } ) = 𝑤 → ( ( ∀ 𝑥 ( 𝑥 ≈ 𝑣 → 𝜑 ) → [ ( ( 𝑤 ∖ { 𝑧 } ) ∪ { 𝑧 } ) / 𝑥 ] 𝜑 ) ↔ ( ∀ 𝑥 ( 𝑥 ≈ 𝑣 → 𝜑 ) → [ 𝑤 / 𝑥 ] 𝜑 ) ) ) |
55 |
52 54
|
imbitrid |
⊢ ( ( ( 𝑤 ∖ { 𝑧 } ) ∪ { 𝑧 } ) = 𝑤 → ( ( 𝑣 ∈ ω ∧ ( 𝑤 ∖ { 𝑧 } ) ≈ 𝑣 ) → ( ∀ 𝑥 ( 𝑥 ≈ 𝑣 → 𝜑 ) → [ 𝑤 / 𝑥 ] 𝜑 ) ) ) |
56 |
24 28 55
|
3syl |
⊢ ( 𝑧 ∈ 𝑤 → ( ( 𝑣 ∈ ω ∧ ( 𝑤 ∖ { 𝑧 } ) ≈ 𝑣 ) → ( ∀ 𝑥 ( 𝑥 ≈ 𝑣 → 𝜑 ) → [ 𝑤 / 𝑥 ] 𝜑 ) ) ) |
57 |
56
|
expd |
⊢ ( 𝑧 ∈ 𝑤 → ( 𝑣 ∈ ω → ( ( 𝑤 ∖ { 𝑧 } ) ≈ 𝑣 → ( ∀ 𝑥 ( 𝑥 ≈ 𝑣 → 𝜑 ) → [ 𝑤 / 𝑥 ] 𝜑 ) ) ) ) |
58 |
57
|
com12 |
⊢ ( 𝑣 ∈ ω → ( 𝑧 ∈ 𝑤 → ( ( 𝑤 ∖ { 𝑧 } ) ≈ 𝑣 → ( ∀ 𝑥 ( 𝑥 ≈ 𝑣 → 𝜑 ) → [ 𝑤 / 𝑥 ] 𝜑 ) ) ) ) |
59 |
58
|
rexlimdv |
⊢ ( 𝑣 ∈ ω → ( ∃ 𝑧 ∈ 𝑤 ( 𝑤 ∖ { 𝑧 } ) ≈ 𝑣 → ( ∀ 𝑥 ( 𝑥 ≈ 𝑣 → 𝜑 ) → [ 𝑤 / 𝑥 ] 𝜑 ) ) ) |
60 |
59
|
adantr |
⊢ ( ( 𝑣 ∈ ω ∧ 𝑤 ≈ suc 𝑣 ) → ( ∃ 𝑧 ∈ 𝑤 ( 𝑤 ∖ { 𝑧 } ) ≈ 𝑣 → ( ∀ 𝑥 ( 𝑥 ≈ 𝑣 → 𝜑 ) → [ 𝑤 / 𝑥 ] 𝜑 ) ) ) |
61 |
23 60
|
mpd |
⊢ ( ( 𝑣 ∈ ω ∧ 𝑤 ≈ suc 𝑣 ) → ( ∀ 𝑥 ( 𝑥 ≈ 𝑣 → 𝜑 ) → [ 𝑤 / 𝑥 ] 𝜑 ) ) |
62 |
61
|
ex |
⊢ ( 𝑣 ∈ ω → ( 𝑤 ≈ suc 𝑣 → ( ∀ 𝑥 ( 𝑥 ≈ 𝑣 → 𝜑 ) → [ 𝑤 / 𝑥 ] 𝜑 ) ) ) |
63 |
62
|
com23 |
⊢ ( 𝑣 ∈ ω → ( ∀ 𝑥 ( 𝑥 ≈ 𝑣 → 𝜑 ) → ( 𝑤 ≈ suc 𝑣 → [ 𝑤 / 𝑥 ] 𝜑 ) ) ) |
64 |
63
|
alrimdv |
⊢ ( 𝑣 ∈ ω → ( ∀ 𝑥 ( 𝑥 ≈ 𝑣 → 𝜑 ) → ∀ 𝑤 ( 𝑤 ≈ suc 𝑣 → [ 𝑤 / 𝑥 ] 𝜑 ) ) ) |
65 |
|
nfv |
⊢ Ⅎ 𝑤 ( 𝑥 ≈ suc 𝑣 → 𝜑 ) |
66 |
|
nfv |
⊢ Ⅎ 𝑥 𝑤 ≈ suc 𝑣 |
67 |
|
nfsbc1v |
⊢ Ⅎ 𝑥 [ 𝑤 / 𝑥 ] 𝜑 |
68 |
66 67
|
nfim |
⊢ Ⅎ 𝑥 ( 𝑤 ≈ suc 𝑣 → [ 𝑤 / 𝑥 ] 𝜑 ) |
69 |
|
breq1 |
⊢ ( 𝑥 = 𝑤 → ( 𝑥 ≈ suc 𝑣 ↔ 𝑤 ≈ suc 𝑣 ) ) |
70 |
|
sbceq1a |
⊢ ( 𝑥 = 𝑤 → ( 𝜑 ↔ [ 𝑤 / 𝑥 ] 𝜑 ) ) |
71 |
69 70
|
imbi12d |
⊢ ( 𝑥 = 𝑤 → ( ( 𝑥 ≈ suc 𝑣 → 𝜑 ) ↔ ( 𝑤 ≈ suc 𝑣 → [ 𝑤 / 𝑥 ] 𝜑 ) ) ) |
72 |
65 68 71
|
cbvalv1 |
⊢ ( ∀ 𝑥 ( 𝑥 ≈ suc 𝑣 → 𝜑 ) ↔ ∀ 𝑤 ( 𝑤 ≈ suc 𝑣 → [ 𝑤 / 𝑥 ] 𝜑 ) ) |
73 |
64 72
|
imbitrrdi |
⊢ ( 𝑣 ∈ ω → ( ∀ 𝑥 ( 𝑥 ≈ 𝑣 → 𝜑 ) → ∀ 𝑥 ( 𝑥 ≈ suc 𝑣 → 𝜑 ) ) ) |
74 |
10 13 16 20 73
|
finds1 |
⊢ ( 𝑤 ∈ ω → ∀ 𝑥 ( 𝑥 ≈ 𝑤 → 𝜑 ) ) |
75 |
74
|
19.21bi |
⊢ ( 𝑤 ∈ ω → ( 𝑥 ≈ 𝑤 → 𝜑 ) ) |
76 |
75
|
rexlimiv |
⊢ ( ∃ 𝑤 ∈ ω 𝑥 ≈ 𝑤 → 𝜑 ) |
77 |
7 76
|
sylbi |
⊢ ( 𝑥 ∈ Fin → 𝜑 ) |
78 |
4 77
|
vtoclga |
⊢ ( 𝐴 ∈ Fin → 𝜏 ) |