Step | Hyp | Ref
| Expression |
1 | | findcard2.4 |
. 2
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) |
2 | | isfi 7865 |
. . 3
⊢ (𝑥 ∈ Fin ↔ ∃𝑤 ∈ ω 𝑥 ≈ 𝑤) |
3 | | breq2 4587 |
. . . . . . . 8
⊢ (𝑤 = ∅ → (𝑥 ≈ 𝑤 ↔ 𝑥 ≈ ∅)) |
4 | 3 | imbi1d 330 |
. . . . . . 7
⊢ (𝑤 = ∅ → ((𝑥 ≈ 𝑤 → 𝜑) ↔ (𝑥 ≈ ∅ → 𝜑))) |
5 | 4 | albidv 1836 |
. . . . . 6
⊢ (𝑤 = ∅ → (∀𝑥(𝑥 ≈ 𝑤 → 𝜑) ↔ ∀𝑥(𝑥 ≈ ∅ → 𝜑))) |
6 | | breq2 4587 |
. . . . . . . 8
⊢ (𝑤 = 𝑣 → (𝑥 ≈ 𝑤 ↔ 𝑥 ≈ 𝑣)) |
7 | 6 | imbi1d 330 |
. . . . . . 7
⊢ (𝑤 = 𝑣 → ((𝑥 ≈ 𝑤 → 𝜑) ↔ (𝑥 ≈ 𝑣 → 𝜑))) |
8 | 7 | albidv 1836 |
. . . . . 6
⊢ (𝑤 = 𝑣 → (∀𝑥(𝑥 ≈ 𝑤 → 𝜑) ↔ ∀𝑥(𝑥 ≈ 𝑣 → 𝜑))) |
9 | | breq2 4587 |
. . . . . . . 8
⊢ (𝑤 = suc 𝑣 → (𝑥 ≈ 𝑤 ↔ 𝑥 ≈ suc 𝑣)) |
10 | 9 | imbi1d 330 |
. . . . . . 7
⊢ (𝑤 = suc 𝑣 → ((𝑥 ≈ 𝑤 → 𝜑) ↔ (𝑥 ≈ suc 𝑣 → 𝜑))) |
11 | 10 | albidv 1836 |
. . . . . 6
⊢ (𝑤 = suc 𝑣 → (∀𝑥(𝑥 ≈ 𝑤 → 𝜑) ↔ ∀𝑥(𝑥 ≈ suc 𝑣 → 𝜑))) |
12 | | en0 7905 |
. . . . . . . 8
⊢ (𝑥 ≈ ∅ ↔ 𝑥 = ∅) |
13 | | findcard2.5 |
. . . . . . . . 9
⊢ 𝜓 |
14 | | findcard2.1 |
. . . . . . . . 9
⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓)) |
15 | 13, 14 | mpbiri 247 |
. . . . . . . 8
⊢ (𝑥 = ∅ → 𝜑) |
16 | 12, 15 | sylbi 206 |
. . . . . . 7
⊢ (𝑥 ≈ ∅ → 𝜑) |
17 | 16 | ax-gen 1713 |
. . . . . 6
⊢
∀𝑥(𝑥 ≈ ∅ → 𝜑) |
18 | | nsuceq0 5722 |
. . . . . . . . . . . 12
⊢ suc 𝑣 ≠ ∅ |
19 | | breq1 4586 |
. . . . . . . . . . . . . . . 16
⊢ (𝑤 = ∅ → (𝑤 ≈ suc 𝑣 ↔ ∅ ≈ suc 𝑣)) |
20 | 19 | anbi2d 736 |
. . . . . . . . . . . . . . 15
⊢ (𝑤 = ∅ → ((𝑣 ∈ ω ∧ 𝑤 ≈ suc 𝑣) ↔ (𝑣 ∈ ω ∧ ∅ ≈ suc
𝑣))) |
21 | | peano1 6977 |
. . . . . . . . . . . . . . . . . 18
⊢ ∅
∈ ω |
22 | | peano2 6978 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑣 ∈ ω → suc 𝑣 ∈
ω) |
23 | | nneneq 8028 |
. . . . . . . . . . . . . . . . . 18
⊢ ((∅
∈ ω ∧ suc 𝑣
∈ ω) → (∅ ≈ suc 𝑣 ↔ ∅ = suc 𝑣)) |
24 | 21, 22, 23 | sylancr 694 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑣 ∈ ω → (∅
≈ suc 𝑣 ↔
∅ = suc 𝑣)) |
25 | 24 | biimpa 500 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑣 ∈ ω ∧ ∅
≈ suc 𝑣) →
∅ = suc 𝑣) |
26 | 25 | eqcomd 2616 |
. . . . . . . . . . . . . . 15
⊢ ((𝑣 ∈ ω ∧ ∅
≈ suc 𝑣) → suc
𝑣 =
∅) |
27 | 20, 26 | syl6bi 242 |
. . . . . . . . . . . . . 14
⊢ (𝑤 = ∅ → ((𝑣 ∈ ω ∧ 𝑤 ≈ suc 𝑣) → suc 𝑣 = ∅)) |
28 | 27 | com12 32 |
. . . . . . . . . . . . 13
⊢ ((𝑣 ∈ ω ∧ 𝑤 ≈ suc 𝑣) → (𝑤 = ∅ → suc 𝑣 = ∅)) |
29 | 28 | necon3d 2803 |
. . . . . . . . . . . 12
⊢ ((𝑣 ∈ ω ∧ 𝑤 ≈ suc 𝑣) → (suc 𝑣 ≠ ∅ → 𝑤 ≠ ∅)) |
30 | 18, 29 | mpi 20 |
. . . . . . . . . . 11
⊢ ((𝑣 ∈ ω ∧ 𝑤 ≈ suc 𝑣) → 𝑤 ≠ ∅) |
31 | 30 | ex 449 |
. . . . . . . . . 10
⊢ (𝑣 ∈ ω → (𝑤 ≈ suc 𝑣 → 𝑤 ≠ ∅)) |
32 | | n0 3890 |
. . . . . . . . . . . 12
⊢ (𝑤 ≠ ∅ ↔
∃𝑧 𝑧 ∈ 𝑤) |
33 | | dif1en 8078 |
. . . . . . . . . . . . . . 15
⊢ ((𝑣 ∈ ω ∧ 𝑤 ≈ suc 𝑣 ∧ 𝑧 ∈ 𝑤) → (𝑤 ∖ {𝑧}) ≈ 𝑣) |
34 | 33 | 3expia 1259 |
. . . . . . . . . . . . . 14
⊢ ((𝑣 ∈ ω ∧ 𝑤 ≈ suc 𝑣) → (𝑧 ∈ 𝑤 → (𝑤 ∖ {𝑧}) ≈ 𝑣)) |
35 | | snssi 4280 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑧 ∈ 𝑤 → {𝑧} ⊆ 𝑤) |
36 | | uncom 3719 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑤 ∖ {𝑧}) ∪ {𝑧}) = ({𝑧} ∪ (𝑤 ∖ {𝑧})) |
37 | | undif 4001 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ({𝑧} ⊆ 𝑤 ↔ ({𝑧} ∪ (𝑤 ∖ {𝑧})) = 𝑤) |
38 | 37 | biimpi 205 |
. . . . . . . . . . . . . . . . . . 19
⊢ ({𝑧} ⊆ 𝑤 → ({𝑧} ∪ (𝑤 ∖ {𝑧})) = 𝑤) |
39 | 36, 38 | syl5eq 2656 |
. . . . . . . . . . . . . . . . . 18
⊢ ({𝑧} ⊆ 𝑤 → ((𝑤 ∖ {𝑧}) ∪ {𝑧}) = 𝑤) |
40 | | vex 3176 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 𝑤 ∈ V |
41 | | difexg 4735 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑤 ∈ V → (𝑤 ∖ {𝑧}) ∈ V) |
42 | 40, 41 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑤 ∖ {𝑧}) ∈ V |
43 | | breq1 4586 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 = (𝑤 ∖ {𝑧}) → (𝑦 ≈ 𝑣 ↔ (𝑤 ∖ {𝑧}) ≈ 𝑣)) |
44 | 43 | anbi2d 736 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 = (𝑤 ∖ {𝑧}) → ((𝑣 ∈ ω ∧ 𝑦 ≈ 𝑣) ↔ (𝑣 ∈ ω ∧ (𝑤 ∖ {𝑧}) ≈ 𝑣))) |
45 | | uneq1 3722 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 = (𝑤 ∖ {𝑧}) → (𝑦 ∪ {𝑧}) = ((𝑤 ∖ {𝑧}) ∪ {𝑧})) |
46 | 45 | sbceq1d 3407 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 = (𝑤 ∖ {𝑧}) → ([(𝑦 ∪ {𝑧}) / 𝑥]𝜑 ↔ [((𝑤 ∖ {𝑧}) ∪ {𝑧}) / 𝑥]𝜑)) |
47 | 46 | imbi2d 329 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 = (𝑤 ∖ {𝑧}) → ((∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [(𝑦 ∪ {𝑧}) / 𝑥]𝜑) ↔ (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [((𝑤 ∖ {𝑧}) ∪ {𝑧}) / 𝑥]𝜑))) |
48 | 44, 47 | imbi12d 333 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (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)))) |
49 | | breq1 4586 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑥 = 𝑦 → (𝑥 ≈ 𝑣 ↔ 𝑦 ≈ 𝑣)) |
50 | | findcard2.2 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) |
51 | 49, 50 | imbi12d 333 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑥 = 𝑦 → ((𝑥 ≈ 𝑣 → 𝜑) ↔ (𝑦 ≈ 𝑣 → 𝜒))) |
52 | 51 | spv 2248 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → (𝑦 ≈ 𝑣 → 𝜒)) |
53 | | rspe 2986 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑣 ∈ ω ∧ 𝑦 ≈ 𝑣) → ∃𝑣 ∈ ω 𝑦 ≈ 𝑣) |
54 | | isfi 7865 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑦 ∈ Fin ↔ ∃𝑣 ∈ ω 𝑦 ≈ 𝑣) |
55 | 53, 54 | sylibr 223 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑣 ∈ ω ∧ 𝑦 ≈ 𝑣) → 𝑦 ∈ Fin) |
56 | | pm2.27 41 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑦 ≈ 𝑣 → ((𝑦 ≈ 𝑣 → 𝜒) → 𝜒)) |
57 | 56 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑣 ∈ ω ∧ 𝑦 ≈ 𝑣) → ((𝑦 ≈ 𝑣 → 𝜒) → 𝜒)) |
58 | | findcard2.6 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 ∈ Fin → (𝜒 → 𝜃)) |
59 | 55, 57, 58 | sylsyld 59 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑣 ∈ ω ∧ 𝑦 ≈ 𝑣) → ((𝑦 ≈ 𝑣 → 𝜒) → 𝜃)) |
60 | 52, 59 | syl5 33 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑣 ∈ ω ∧ 𝑦 ≈ 𝑣) → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → 𝜃)) |
61 | | vex 3176 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ 𝑦 ∈ V |
62 | | snex 4835 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ {𝑧} ∈ V |
63 | 61, 62 | unex 6854 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 ∪ {𝑧}) ∈ V |
64 | | findcard2.3 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝜑 ↔ 𝜃)) |
65 | 63, 64 | sbcie 3437 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
([(𝑦 ∪
{𝑧}) / 𝑥]𝜑 ↔ 𝜃) |
66 | 60, 65 | syl6ibr 241 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑣 ∈ ω ∧ 𝑦 ≈ 𝑣) → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [(𝑦 ∪ {𝑧}) / 𝑥]𝜑)) |
67 | 42, 48, 66 | vtocl 3232 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑣 ∈ ω ∧ (𝑤 ∖ {𝑧}) ≈ 𝑣) → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [((𝑤 ∖ {𝑧}) ∪ {𝑧}) / 𝑥]𝜑)) |
68 | | dfsbcq 3404 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑤 ∖ {𝑧}) ∪ {𝑧}) = 𝑤 → ([((𝑤 ∖ {𝑧}) ∪ {𝑧}) / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝜑)) |
69 | 68 | imbi2d 329 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑤 ∖ {𝑧}) ∪ {𝑧}) = 𝑤 → ((∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [((𝑤 ∖ {𝑧}) ∪ {𝑧}) / 𝑥]𝜑) ↔ (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑))) |
70 | 67, 69 | syl5ib 233 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑤 ∖ {𝑧}) ∪ {𝑧}) = 𝑤 → ((𝑣 ∈ ω ∧ (𝑤 ∖ {𝑧}) ≈ 𝑣) → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑))) |
71 | 35, 39, 70 | 3syl 18 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑧 ∈ 𝑤 → ((𝑣 ∈ ω ∧ (𝑤 ∖ {𝑧}) ≈ 𝑣) → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑))) |
72 | 71 | expd 451 |
. . . . . . . . . . . . . . . 16
⊢ (𝑧 ∈ 𝑤 → (𝑣 ∈ ω → ((𝑤 ∖ {𝑧}) ≈ 𝑣 → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑)))) |
73 | 72 | com12 32 |
. . . . . . . . . . . . . . 15
⊢ (𝑣 ∈ ω → (𝑧 ∈ 𝑤 → ((𝑤 ∖ {𝑧}) ≈ 𝑣 → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑)))) |
74 | 73 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝑣 ∈ ω ∧ 𝑤 ≈ suc 𝑣) → (𝑧 ∈ 𝑤 → ((𝑤 ∖ {𝑧}) ≈ 𝑣 → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑)))) |
75 | 34, 74 | mpdd 42 |
. . . . . . . . . . . . 13
⊢ ((𝑣 ∈ ω ∧ 𝑤 ≈ suc 𝑣) → (𝑧 ∈ 𝑤 → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑))) |
76 | 75 | exlimdv 1848 |
. . . . . . . . . . . 12
⊢ ((𝑣 ∈ ω ∧ 𝑤 ≈ suc 𝑣) → (∃𝑧 𝑧 ∈ 𝑤 → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑))) |
77 | 32, 76 | syl5bi 231 |
. . . . . . . . . . 11
⊢ ((𝑣 ∈ ω ∧ 𝑤 ≈ suc 𝑣) → (𝑤 ≠ ∅ → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑))) |
78 | 77 | ex 449 |
. . . . . . . . . 10
⊢ (𝑣 ∈ ω → (𝑤 ≈ suc 𝑣 → (𝑤 ≠ ∅ → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑)))) |
79 | 31, 78 | mpdd 42 |
. . . . . . . . 9
⊢ (𝑣 ∈ ω → (𝑤 ≈ suc 𝑣 → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑))) |
80 | 79 | com23 84 |
. . . . . . . 8
⊢ (𝑣 ∈ ω →
(∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → (𝑤 ≈ suc 𝑣 → [𝑤 / 𝑥]𝜑))) |
81 | 80 | alrimdv 1844 |
. . . . . . 7
⊢ (𝑣 ∈ ω →
(∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → ∀𝑤(𝑤 ≈ suc 𝑣 → [𝑤 / 𝑥]𝜑))) |
82 | | nfv 1830 |
. . . . . . . 8
⊢
Ⅎ𝑤(𝑥 ≈ suc 𝑣 → 𝜑) |
83 | | nfv 1830 |
. . . . . . . . 9
⊢
Ⅎ𝑥 𝑤 ≈ suc 𝑣 |
84 | | nfsbc1v 3422 |
. . . . . . . . 9
⊢
Ⅎ𝑥[𝑤 / 𝑥]𝜑 |
85 | 83, 84 | nfim 1813 |
. . . . . . . 8
⊢
Ⅎ𝑥(𝑤 ≈ suc 𝑣 → [𝑤 / 𝑥]𝜑) |
86 | | breq1 4586 |
. . . . . . . . 9
⊢ (𝑥 = 𝑤 → (𝑥 ≈ suc 𝑣 ↔ 𝑤 ≈ suc 𝑣)) |
87 | | sbceq1a 3413 |
. . . . . . . . 9
⊢ (𝑥 = 𝑤 → (𝜑 ↔ [𝑤 / 𝑥]𝜑)) |
88 | 86, 87 | imbi12d 333 |
. . . . . . . 8
⊢ (𝑥 = 𝑤 → ((𝑥 ≈ suc 𝑣 → 𝜑) ↔ (𝑤 ≈ suc 𝑣 → [𝑤 / 𝑥]𝜑))) |
89 | 82, 85, 88 | cbval 2259 |
. . . . . . 7
⊢
(∀𝑥(𝑥 ≈ suc 𝑣 → 𝜑) ↔ ∀𝑤(𝑤 ≈ suc 𝑣 → [𝑤 / 𝑥]𝜑)) |
90 | 81, 89 | syl6ibr 241 |
. . . . . 6
⊢ (𝑣 ∈ ω →
(∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → ∀𝑥(𝑥 ≈ suc 𝑣 → 𝜑))) |
91 | 5, 8, 11, 17, 90 | finds1 6987 |
. . . . 5
⊢ (𝑤 ∈ ω →
∀𝑥(𝑥 ≈ 𝑤 → 𝜑)) |
92 | 91 | 19.21bi 2047 |
. . . 4
⊢ (𝑤 ∈ ω → (𝑥 ≈ 𝑤 → 𝜑)) |
93 | 92 | rexlimiv 3009 |
. . 3
⊢
(∃𝑤 ∈
ω 𝑥 ≈ 𝑤 → 𝜑) |
94 | 2, 93 | sylbi 206 |
. 2
⊢ (𝑥 ∈ Fin → 𝜑) |
95 | 1, 94 | vtoclga 3245 |
1
⊢ (𝐴 ∈ Fin → 𝜏) |