Step | Hyp | Ref
| Expression |
1 | | findcard2.4 |
. 2
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) |
2 | | isfi 8652 |
. . 3
⊢ (𝑥 ∈ Fin ↔ ∃𝑤 ∈ ω 𝑥 ≈ 𝑤) |
3 | | breq2 5057 |
. . . . . . . 8
⊢ (𝑤 = ∅ → (𝑥 ≈ 𝑤 ↔ 𝑥 ≈ ∅)) |
4 | 3 | imbi1d 345 |
. . . . . . 7
⊢ (𝑤 = ∅ → ((𝑥 ≈ 𝑤 → 𝜑) ↔ (𝑥 ≈ ∅ → 𝜑))) |
5 | 4 | albidv 1928 |
. . . . . 6
⊢ (𝑤 = ∅ → (∀𝑥(𝑥 ≈ 𝑤 → 𝜑) ↔ ∀𝑥(𝑥 ≈ ∅ → 𝜑))) |
6 | | breq2 5057 |
. . . . . . . 8
⊢ (𝑤 = 𝑣 → (𝑥 ≈ 𝑤 ↔ 𝑥 ≈ 𝑣)) |
7 | 6 | imbi1d 345 |
. . . . . . 7
⊢ (𝑤 = 𝑣 → ((𝑥 ≈ 𝑤 → 𝜑) ↔ (𝑥 ≈ 𝑣 → 𝜑))) |
8 | 7 | albidv 1928 |
. . . . . 6
⊢ (𝑤 = 𝑣 → (∀𝑥(𝑥 ≈ 𝑤 → 𝜑) ↔ ∀𝑥(𝑥 ≈ 𝑣 → 𝜑))) |
9 | | breq2 5057 |
. . . . . . . 8
⊢ (𝑤 = suc 𝑣 → (𝑥 ≈ 𝑤 ↔ 𝑥 ≈ suc 𝑣)) |
10 | 9 | imbi1d 345 |
. . . . . . 7
⊢ (𝑤 = suc 𝑣 → ((𝑥 ≈ 𝑤 → 𝜑) ↔ (𝑥 ≈ suc 𝑣 → 𝜑))) |
11 | 10 | albidv 1928 |
. . . . . 6
⊢ (𝑤 = suc 𝑣 → (∀𝑥(𝑥 ≈ 𝑤 → 𝜑) ↔ ∀𝑥(𝑥 ≈ suc 𝑣 → 𝜑))) |
12 | | en0 8691 |
. . . . . . . 8
⊢ (𝑥 ≈ ∅ ↔ 𝑥 = ∅) |
13 | | findcard2.5 |
. . . . . . . . 9
⊢ 𝜓 |
14 | | findcard2.1 |
. . . . . . . . 9
⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓)) |
15 | 13, 14 | mpbiri 261 |
. . . . . . . 8
⊢ (𝑥 = ∅ → 𝜑) |
16 | 12, 15 | sylbi 220 |
. . . . . . 7
⊢ (𝑥 ≈ ∅ → 𝜑) |
17 | 16 | ax-gen 1803 |
. . . . . 6
⊢
∀𝑥(𝑥 ≈ ∅ → 𝜑) |
18 | | rexdif1en 8839 |
. . . . . . . . . . 11
⊢ ((𝑣 ∈ ω ∧ 𝑤 ≈ suc 𝑣) → ∃𝑧 ∈ 𝑤 (𝑤 ∖ {𝑧}) ≈ 𝑣) |
19 | | snssi 4721 |
. . . . . . . . . . . . . . . 16
⊢ (𝑧 ∈ 𝑤 → {𝑧} ⊆ 𝑤) |
20 | | uncom 4067 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑤 ∖ {𝑧}) ∪ {𝑧}) = ({𝑧} ∪ (𝑤 ∖ {𝑧})) |
21 | | undif 4396 |
. . . . . . . . . . . . . . . . . 18
⊢ ({𝑧} ⊆ 𝑤 ↔ ({𝑧} ∪ (𝑤 ∖ {𝑧})) = 𝑤) |
22 | 21 | biimpi 219 |
. . . . . . . . . . . . . . . . 17
⊢ ({𝑧} ⊆ 𝑤 → ({𝑧} ∪ (𝑤 ∖ {𝑧})) = 𝑤) |
23 | 20, 22 | eqtrid 2789 |
. . . . . . . . . . . . . . . 16
⊢ ({𝑧} ⊆ 𝑤 → ((𝑤 ∖ {𝑧}) ∪ {𝑧}) = 𝑤) |
24 | | vex 3412 |
. . . . . . . . . . . . . . . . . . 19
⊢ 𝑤 ∈ V |
25 | 24 | difexi 5221 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑤 ∖ {𝑧}) ∈ V |
26 | | breq1 5056 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 = (𝑤 ∖ {𝑧}) → (𝑦 ≈ 𝑣 ↔ (𝑤 ∖ {𝑧}) ≈ 𝑣)) |
27 | 26 | anbi2d 632 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 = (𝑤 ∖ {𝑧}) → ((𝑣 ∈ ω ∧ 𝑦 ≈ 𝑣) ↔ (𝑣 ∈ ω ∧ (𝑤 ∖ {𝑧}) ≈ 𝑣))) |
28 | | uneq1 4070 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 = (𝑤 ∖ {𝑧}) → (𝑦 ∪ {𝑧}) = ((𝑤 ∖ {𝑧}) ∪ {𝑧})) |
29 | 28 | sbceq1d 3699 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 = (𝑤 ∖ {𝑧}) → ([(𝑦 ∪ {𝑧}) / 𝑥]𝜑 ↔ [((𝑤 ∖ {𝑧}) ∪ {𝑧}) / 𝑥]𝜑)) |
30 | 29 | imbi2d 344 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 = (𝑤 ∖ {𝑧}) → ((∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [(𝑦 ∪ {𝑧}) / 𝑥]𝜑) ↔ (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [((𝑤 ∖ {𝑧}) ∪ {𝑧}) / 𝑥]𝜑))) |
31 | 27, 30 | imbi12d 348 |
. . . . . . . . . . . . . . . . . 18
⊢ (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)))) |
32 | | breq1 5056 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 = 𝑦 → (𝑥 ≈ 𝑣 ↔ 𝑦 ≈ 𝑣)) |
33 | | findcard2.2 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) |
34 | 32, 33 | imbi12d 348 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 = 𝑦 → ((𝑥 ≈ 𝑣 → 𝜑) ↔ (𝑦 ≈ 𝑣 → 𝜒))) |
35 | 34 | spvv 2005 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → (𝑦 ≈ 𝑣 → 𝜒)) |
36 | | rspe 3223 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑣 ∈ ω ∧ 𝑦 ≈ 𝑣) → ∃𝑣 ∈ ω 𝑦 ≈ 𝑣) |
37 | | isfi 8652 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 ∈ Fin ↔ ∃𝑣 ∈ ω 𝑦 ≈ 𝑣) |
38 | 36, 37 | sylibr 237 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑣 ∈ ω ∧ 𝑦 ≈ 𝑣) → 𝑦 ∈ Fin) |
39 | | pm2.27 42 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 ≈ 𝑣 → ((𝑦 ≈ 𝑣 → 𝜒) → 𝜒)) |
40 | 39 | adantl 485 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑣 ∈ ω ∧ 𝑦 ≈ 𝑣) → ((𝑦 ≈ 𝑣 → 𝜒) → 𝜒)) |
41 | | findcard2.6 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 ∈ Fin → (𝜒 → 𝜃)) |
42 | 38, 40, 41 | sylsyld 61 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑣 ∈ ω ∧ 𝑦 ≈ 𝑣) → ((𝑦 ≈ 𝑣 → 𝜒) → 𝜃)) |
43 | 35, 42 | syl5 34 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑣 ∈ ω ∧ 𝑦 ≈ 𝑣) → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → 𝜃)) |
44 | | vex 3412 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 𝑦 ∈ V |
45 | | snex 5324 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ {𝑧} ∈ V |
46 | 44, 45 | unex 7531 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 ∪ {𝑧}) ∈ V |
47 | | findcard2.3 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝜑 ↔ 𝜃)) |
48 | 46, 47 | sbcie 3737 |
. . . . . . . . . . . . . . . . . . 19
⊢
([(𝑦 ∪
{𝑧}) / 𝑥]𝜑 ↔ 𝜃) |
49 | 43, 48 | syl6ibr 255 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑣 ∈ ω ∧ 𝑦 ≈ 𝑣) → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [(𝑦 ∪ {𝑧}) / 𝑥]𝜑)) |
50 | 25, 31, 49 | vtocl 3474 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑣 ∈ ω ∧ (𝑤 ∖ {𝑧}) ≈ 𝑣) → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [((𝑤 ∖ {𝑧}) ∪ {𝑧}) / 𝑥]𝜑)) |
51 | | dfsbcq 3696 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑤 ∖ {𝑧}) ∪ {𝑧}) = 𝑤 → ([((𝑤 ∖ {𝑧}) ∪ {𝑧}) / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝜑)) |
52 | 51 | imbi2d 344 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑤 ∖ {𝑧}) ∪ {𝑧}) = 𝑤 → ((∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [((𝑤 ∖ {𝑧}) ∪ {𝑧}) / 𝑥]𝜑) ↔ (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑))) |
53 | 50, 52 | syl5ib 247 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑤 ∖ {𝑧}) ∪ {𝑧}) = 𝑤 → ((𝑣 ∈ ω ∧ (𝑤 ∖ {𝑧}) ≈ 𝑣) → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑))) |
54 | 19, 23, 53 | 3syl 18 |
. . . . . . . . . . . . . . 15
⊢ (𝑧 ∈ 𝑤 → ((𝑣 ∈ ω ∧ (𝑤 ∖ {𝑧}) ≈ 𝑣) → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑))) |
55 | 54 | expd 419 |
. . . . . . . . . . . . . 14
⊢ (𝑧 ∈ 𝑤 → (𝑣 ∈ ω → ((𝑤 ∖ {𝑧}) ≈ 𝑣 → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑)))) |
56 | 55 | com12 32 |
. . . . . . . . . . . . 13
⊢ (𝑣 ∈ ω → (𝑧 ∈ 𝑤 → ((𝑤 ∖ {𝑧}) ≈ 𝑣 → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑)))) |
57 | 56 | rexlimdv 3202 |
. . . . . . . . . . . 12
⊢ (𝑣 ∈ ω →
(∃𝑧 ∈ 𝑤 (𝑤 ∖ {𝑧}) ≈ 𝑣 → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑))) |
58 | 57 | adantr 484 |
. . . . . . . . . . 11
⊢ ((𝑣 ∈ ω ∧ 𝑤 ≈ suc 𝑣) → (∃𝑧 ∈ 𝑤 (𝑤 ∖ {𝑧}) ≈ 𝑣 → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑))) |
59 | 18, 58 | mpd 15 |
. . . . . . . . . 10
⊢ ((𝑣 ∈ ω ∧ 𝑤 ≈ suc 𝑣) → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑)) |
60 | 59 | ex 416 |
. . . . . . . . 9
⊢ (𝑣 ∈ ω → (𝑤 ≈ suc 𝑣 → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑))) |
61 | 60 | com23 86 |
. . . . . . . 8
⊢ (𝑣 ∈ ω →
(∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → (𝑤 ≈ suc 𝑣 → [𝑤 / 𝑥]𝜑))) |
62 | 61 | alrimdv 1937 |
. . . . . . 7
⊢ (𝑣 ∈ ω →
(∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → ∀𝑤(𝑤 ≈ suc 𝑣 → [𝑤 / 𝑥]𝜑))) |
63 | | nfv 1922 |
. . . . . . . 8
⊢
Ⅎ𝑤(𝑥 ≈ suc 𝑣 → 𝜑) |
64 | | nfv 1922 |
. . . . . . . . 9
⊢
Ⅎ𝑥 𝑤 ≈ suc 𝑣 |
65 | | nfsbc1v 3714 |
. . . . . . . . 9
⊢
Ⅎ𝑥[𝑤 / 𝑥]𝜑 |
66 | 64, 65 | nfim 1904 |
. . . . . . . 8
⊢
Ⅎ𝑥(𝑤 ≈ suc 𝑣 → [𝑤 / 𝑥]𝜑) |
67 | | breq1 5056 |
. . . . . . . . 9
⊢ (𝑥 = 𝑤 → (𝑥 ≈ suc 𝑣 ↔ 𝑤 ≈ suc 𝑣)) |
68 | | sbceq1a 3705 |
. . . . . . . . 9
⊢ (𝑥 = 𝑤 → (𝜑 ↔ [𝑤 / 𝑥]𝜑)) |
69 | 67, 68 | imbi12d 348 |
. . . . . . . 8
⊢ (𝑥 = 𝑤 → ((𝑥 ≈ suc 𝑣 → 𝜑) ↔ (𝑤 ≈ suc 𝑣 → [𝑤 / 𝑥]𝜑))) |
70 | 63, 66, 69 | cbvalv1 2341 |
. . . . . . 7
⊢
(∀𝑥(𝑥 ≈ suc 𝑣 → 𝜑) ↔ ∀𝑤(𝑤 ≈ suc 𝑣 → [𝑤 / 𝑥]𝜑)) |
71 | 62, 70 | syl6ibr 255 |
. . . . . 6
⊢ (𝑣 ∈ ω →
(∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → ∀𝑥(𝑥 ≈ suc 𝑣 → 𝜑))) |
72 | 5, 8, 11, 17, 71 | finds1 7679 |
. . . . 5
⊢ (𝑤 ∈ ω →
∀𝑥(𝑥 ≈ 𝑤 → 𝜑)) |
73 | 72 | 19.21bi 2186 |
. . . 4
⊢ (𝑤 ∈ ω → (𝑥 ≈ 𝑤 → 𝜑)) |
74 | 73 | rexlimiv 3199 |
. . 3
⊢
(∃𝑤 ∈
ω 𝑥 ≈ 𝑤 → 𝜑) |
75 | 2, 74 | sylbi 220 |
. 2
⊢ (𝑥 ∈ Fin → 𝜑) |
76 | 1, 75 | vtoclga 3489 |
1
⊢ (𝐴 ∈ Fin → 𝜏) |