Step | Hyp | Ref
| Expression |
1 | | findcard2OLD.4 |
. 2
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) |
2 | | isfi 8579 |
. . 3
⊢ (𝑥 ∈ Fin ↔ ∃𝑤 ∈ ω 𝑥 ≈ 𝑤) |
3 | | breq2 5034 |
. . . . . . . 8
⊢ (𝑤 = ∅ → (𝑥 ≈ 𝑤 ↔ 𝑥 ≈ ∅)) |
4 | 3 | imbi1d 345 |
. . . . . . 7
⊢ (𝑤 = ∅ → ((𝑥 ≈ 𝑤 → 𝜑) ↔ (𝑥 ≈ ∅ → 𝜑))) |
5 | 4 | albidv 1927 |
. . . . . 6
⊢ (𝑤 = ∅ → (∀𝑥(𝑥 ≈ 𝑤 → 𝜑) ↔ ∀𝑥(𝑥 ≈ ∅ → 𝜑))) |
6 | | breq2 5034 |
. . . . . . . 8
⊢ (𝑤 = 𝑣 → (𝑥 ≈ 𝑤 ↔ 𝑥 ≈ 𝑣)) |
7 | 6 | imbi1d 345 |
. . . . . . 7
⊢ (𝑤 = 𝑣 → ((𝑥 ≈ 𝑤 → 𝜑) ↔ (𝑥 ≈ 𝑣 → 𝜑))) |
8 | 7 | albidv 1927 |
. . . . . 6
⊢ (𝑤 = 𝑣 → (∀𝑥(𝑥 ≈ 𝑤 → 𝜑) ↔ ∀𝑥(𝑥 ≈ 𝑣 → 𝜑))) |
9 | | breq2 5034 |
. . . . . . . 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 1927 |
. . . . . 6
⊢ (𝑤 = suc 𝑣 → (∀𝑥(𝑥 ≈ 𝑤 → 𝜑) ↔ ∀𝑥(𝑥 ≈ suc 𝑣 → 𝜑))) |
12 | | en0 8618 |
. . . . . . . 8
⊢ (𝑥 ≈ ∅ ↔ 𝑥 = ∅) |
13 | | findcard2OLD.5 |
. . . . . . . . 9
⊢ 𝜓 |
14 | | findcard2OLD.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 1802 |
. . . . . 6
⊢
∀𝑥(𝑥 ≈ ∅ → 𝜑) |
18 | | nsuceq0 6252 |
. . . . . . . . . . . 12
⊢ suc 𝑣 ≠ ∅ |
19 | | breq1 5033 |
. . . . . . . . . . . . . . . 16
⊢ (𝑤 = ∅ → (𝑤 ≈ suc 𝑣 ↔ ∅ ≈ suc 𝑣)) |
20 | 19 | anbi2d 632 |
. . . . . . . . . . . . . . 15
⊢ (𝑤 = ∅ → ((𝑣 ∈ ω ∧ 𝑤 ≈ suc 𝑣) ↔ (𝑣 ∈ ω ∧ ∅ ≈ suc
𝑣))) |
21 | | peano1 7620 |
. . . . . . . . . . . . . . . . . 18
⊢ ∅
∈ ω |
22 | | peano2 7621 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑣 ∈ ω → suc 𝑣 ∈
ω) |
23 | | nneneq 8750 |
. . . . . . . . . . . . . . . . . 18
⊢ ((∅
∈ ω ∧ suc 𝑣
∈ ω) → (∅ ≈ suc 𝑣 ↔ ∅ = suc 𝑣)) |
24 | 21, 22, 23 | sylancr 590 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑣 ∈ ω → (∅
≈ suc 𝑣 ↔
∅ = suc 𝑣)) |
25 | 24 | biimpa 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑣 ∈ ω ∧ ∅
≈ suc 𝑣) →
∅ = suc 𝑣) |
26 | 25 | eqcomd 2744 |
. . . . . . . . . . . . . . 15
⊢ ((𝑣 ∈ ω ∧ ∅
≈ suc 𝑣) → suc
𝑣 =
∅) |
27 | 20, 26 | syl6bi 256 |
. . . . . . . . . . . . . 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 2955 |
. . . . . . . . . . . 12
⊢ ((𝑣 ∈ ω ∧ 𝑤 ≈ suc 𝑣) → (suc 𝑣 ≠ ∅ → 𝑤 ≠ ∅)) |
30 | 18, 29 | mpi 20 |
. . . . . . . . . . 11
⊢ ((𝑣 ∈ ω ∧ 𝑤 ≈ suc 𝑣) → 𝑤 ≠ ∅) |
31 | 30 | ex 416 |
. . . . . . . . . 10
⊢ (𝑣 ∈ ω → (𝑤 ≈ suc 𝑣 → 𝑤 ≠ ∅)) |
32 | | n0 4235 |
. . . . . . . . . . . 12
⊢ (𝑤 ≠ ∅ ↔
∃𝑧 𝑧 ∈ 𝑤) |
33 | | dif1en 8761 |
. . . . . . . . . . . . . . 15
⊢ ((𝑣 ∈ ω ∧ 𝑤 ≈ suc 𝑣 ∧ 𝑧 ∈ 𝑤) → (𝑤 ∖ {𝑧}) ≈ 𝑣) |
34 | 33 | 3expia 1122 |
. . . . . . . . . . . . . 14
⊢ ((𝑣 ∈ ω ∧ 𝑤 ≈ suc 𝑣) → (𝑧 ∈ 𝑤 → (𝑤 ∖ {𝑧}) ≈ 𝑣)) |
35 | | snssi 4696 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑧 ∈ 𝑤 → {𝑧} ⊆ 𝑤) |
36 | | uncom 4043 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑤 ∖ {𝑧}) ∪ {𝑧}) = ({𝑧} ∪ (𝑤 ∖ {𝑧})) |
37 | | undif 4371 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ({𝑧} ⊆ 𝑤 ↔ ({𝑧} ∪ (𝑤 ∖ {𝑧})) = 𝑤) |
38 | 37 | biimpi 219 |
. . . . . . . . . . . . . . . . . . 19
⊢ ({𝑧} ⊆ 𝑤 → ({𝑧} ∪ (𝑤 ∖ {𝑧})) = 𝑤) |
39 | 36, 38 | syl5eq 2785 |
. . . . . . . . . . . . . . . . . 18
⊢ ({𝑧} ⊆ 𝑤 → ((𝑤 ∖ {𝑧}) ∪ {𝑧}) = 𝑤) |
40 | | vex 3402 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 𝑤 ∈ V |
41 | 40 | difexi 5196 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑤 ∖ {𝑧}) ∈ V |
42 | | breq1 5033 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 = (𝑤 ∖ {𝑧}) → (𝑦 ≈ 𝑣 ↔ (𝑤 ∖ {𝑧}) ≈ 𝑣)) |
43 | 42 | anbi2d 632 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 = (𝑤 ∖ {𝑧}) → ((𝑣 ∈ ω ∧ 𝑦 ≈ 𝑣) ↔ (𝑣 ∈ ω ∧ (𝑤 ∖ {𝑧}) ≈ 𝑣))) |
44 | | uneq1 4046 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 = (𝑤 ∖ {𝑧}) → (𝑦 ∪ {𝑧}) = ((𝑤 ∖ {𝑧}) ∪ {𝑧})) |
45 | 44 | sbceq1d 3685 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 = (𝑤 ∖ {𝑧}) → ([(𝑦 ∪ {𝑧}) / 𝑥]𝜑 ↔ [((𝑤 ∖ {𝑧}) ∪ {𝑧}) / 𝑥]𝜑)) |
46 | 45 | imbi2d 344 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 = (𝑤 ∖ {𝑧}) → ((∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [(𝑦 ∪ {𝑧}) / 𝑥]𝜑) ↔ (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [((𝑤 ∖ {𝑧}) ∪ {𝑧}) / 𝑥]𝜑))) |
47 | 43, 46 | imbi12d 348 |
. . . . . . . . . . . . . . . . . . . 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)))) |
48 | | breq1 5033 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑥 = 𝑦 → (𝑥 ≈ 𝑣 ↔ 𝑦 ≈ 𝑣)) |
49 | | findcard2OLD.2 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) |
50 | 48, 49 | imbi12d 348 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑥 = 𝑦 → ((𝑥 ≈ 𝑣 → 𝜑) ↔ (𝑦 ≈ 𝑣 → 𝜒))) |
51 | 50 | spvv 2008 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → (𝑦 ≈ 𝑣 → 𝜒)) |
52 | | rspe 3214 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑣 ∈ ω ∧ 𝑦 ≈ 𝑣) → ∃𝑣 ∈ ω 𝑦 ≈ 𝑣) |
53 | | isfi 8579 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑦 ∈ Fin ↔ ∃𝑣 ∈ ω 𝑦 ≈ 𝑣) |
54 | 52, 53 | sylibr 237 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑣 ∈ ω ∧ 𝑦 ≈ 𝑣) → 𝑦 ∈ Fin) |
55 | | pm2.27 42 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑦 ≈ 𝑣 → ((𝑦 ≈ 𝑣 → 𝜒) → 𝜒)) |
56 | 55 | adantl 485 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑣 ∈ ω ∧ 𝑦 ≈ 𝑣) → ((𝑦 ≈ 𝑣 → 𝜒) → 𝜒)) |
57 | | findcard2OLD.6 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 ∈ Fin → (𝜒 → 𝜃)) |
58 | 54, 56, 57 | sylsyld 61 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑣 ∈ ω ∧ 𝑦 ≈ 𝑣) → ((𝑦 ≈ 𝑣 → 𝜒) → 𝜃)) |
59 | 51, 58 | syl5 34 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑣 ∈ ω ∧ 𝑦 ≈ 𝑣) → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → 𝜃)) |
60 | | vex 3402 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ 𝑦 ∈ V |
61 | | snex 5298 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ {𝑧} ∈ V |
62 | 60, 61 | unex 7487 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 ∪ {𝑧}) ∈ V |
63 | | findcard2OLD.3 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝜑 ↔ 𝜃)) |
64 | 62, 63 | sbcie 3722 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
([(𝑦 ∪
{𝑧}) / 𝑥]𝜑 ↔ 𝜃) |
65 | 59, 64 | syl6ibr 255 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑣 ∈ ω ∧ 𝑦 ≈ 𝑣) → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [(𝑦 ∪ {𝑧}) / 𝑥]𝜑)) |
66 | 41, 47, 65 | vtocl 3463 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑣 ∈ ω ∧ (𝑤 ∖ {𝑧}) ≈ 𝑣) → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [((𝑤 ∖ {𝑧}) ∪ {𝑧}) / 𝑥]𝜑)) |
67 | | dfsbcq 3682 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑤 ∖ {𝑧}) ∪ {𝑧}) = 𝑤 → ([((𝑤 ∖ {𝑧}) ∪ {𝑧}) / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝜑)) |
68 | 67 | imbi2d 344 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑤 ∖ {𝑧}) ∪ {𝑧}) = 𝑤 → ((∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [((𝑤 ∖ {𝑧}) ∪ {𝑧}) / 𝑥]𝜑) ↔ (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑))) |
69 | 66, 68 | syl5ib 247 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑤 ∖ {𝑧}) ∪ {𝑧}) = 𝑤 → ((𝑣 ∈ ω ∧ (𝑤 ∖ {𝑧}) ≈ 𝑣) → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑))) |
70 | 35, 39, 69 | 3syl 18 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑧 ∈ 𝑤 → ((𝑣 ∈ ω ∧ (𝑤 ∖ {𝑧}) ≈ 𝑣) → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑))) |
71 | 70 | expd 419 |
. . . . . . . . . . . . . . . 16
⊢ (𝑧 ∈ 𝑤 → (𝑣 ∈ ω → ((𝑤 ∖ {𝑧}) ≈ 𝑣 → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑)))) |
72 | 71 | com12 32 |
. . . . . . . . . . . . . . 15
⊢ (𝑣 ∈ ω → (𝑧 ∈ 𝑤 → ((𝑤 ∖ {𝑧}) ≈ 𝑣 → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑)))) |
73 | 72 | adantr 484 |
. . . . . . . . . . . . . 14
⊢ ((𝑣 ∈ ω ∧ 𝑤 ≈ suc 𝑣) → (𝑧 ∈ 𝑤 → ((𝑤 ∖ {𝑧}) ≈ 𝑣 → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑)))) |
74 | 34, 73 | mpdd 43 |
. . . . . . . . . . . . 13
⊢ ((𝑣 ∈ ω ∧ 𝑤 ≈ suc 𝑣) → (𝑧 ∈ 𝑤 → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑))) |
75 | 74 | exlimdv 1940 |
. . . . . . . . . . . 12
⊢ ((𝑣 ∈ ω ∧ 𝑤 ≈ suc 𝑣) → (∃𝑧 𝑧 ∈ 𝑤 → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑))) |
76 | 32, 75 | syl5bi 245 |
. . . . . . . . . . 11
⊢ ((𝑣 ∈ ω ∧ 𝑤 ≈ suc 𝑣) → (𝑤 ≠ ∅ → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑))) |
77 | 76 | ex 416 |
. . . . . . . . . 10
⊢ (𝑣 ∈ ω → (𝑤 ≈ suc 𝑣 → (𝑤 ≠ ∅ → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑)))) |
78 | 31, 77 | mpdd 43 |
. . . . . . . . 9
⊢ (𝑣 ∈ ω → (𝑤 ≈ suc 𝑣 → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑))) |
79 | 78 | com23 86 |
. . . . . . . 8
⊢ (𝑣 ∈ ω →
(∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → (𝑤 ≈ suc 𝑣 → [𝑤 / 𝑥]𝜑))) |
80 | 79 | alrimdv 1936 |
. . . . . . 7
⊢ (𝑣 ∈ ω →
(∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → ∀𝑤(𝑤 ≈ suc 𝑣 → [𝑤 / 𝑥]𝜑))) |
81 | | nfv 1921 |
. . . . . . . 8
⊢
Ⅎ𝑤(𝑥 ≈ suc 𝑣 → 𝜑) |
82 | | nfv 1921 |
. . . . . . . . 9
⊢
Ⅎ𝑥 𝑤 ≈ suc 𝑣 |
83 | | nfsbc1v 3700 |
. . . . . . . . 9
⊢
Ⅎ𝑥[𝑤 / 𝑥]𝜑 |
84 | 82, 83 | nfim 1903 |
. . . . . . . 8
⊢
Ⅎ𝑥(𝑤 ≈ suc 𝑣 → [𝑤 / 𝑥]𝜑) |
85 | | breq1 5033 |
. . . . . . . . 9
⊢ (𝑥 = 𝑤 → (𝑥 ≈ suc 𝑣 ↔ 𝑤 ≈ suc 𝑣)) |
86 | | sbceq1a 3691 |
. . . . . . . . 9
⊢ (𝑥 = 𝑤 → (𝜑 ↔ [𝑤 / 𝑥]𝜑)) |
87 | 85, 86 | imbi12d 348 |
. . . . . . . 8
⊢ (𝑥 = 𝑤 → ((𝑥 ≈ suc 𝑣 → 𝜑) ↔ (𝑤 ≈ suc 𝑣 → [𝑤 / 𝑥]𝜑))) |
88 | 81, 84, 87 | cbvalv1 2343 |
. . . . . . 7
⊢
(∀𝑥(𝑥 ≈ suc 𝑣 → 𝜑) ↔ ∀𝑤(𝑤 ≈ suc 𝑣 → [𝑤 / 𝑥]𝜑)) |
89 | 80, 88 | syl6ibr 255 |
. . . . . 6
⊢ (𝑣 ∈ ω →
(∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → ∀𝑥(𝑥 ≈ suc 𝑣 → 𝜑))) |
90 | 5, 8, 11, 17, 89 | finds1 7632 |
. . . . 5
⊢ (𝑤 ∈ ω →
∀𝑥(𝑥 ≈ 𝑤 → 𝜑)) |
91 | 90 | 19.21bi 2190 |
. . . 4
⊢ (𝑤 ∈ ω → (𝑥 ≈ 𝑤 → 𝜑)) |
92 | 91 | rexlimiv 3190 |
. . 3
⊢
(∃𝑤 ∈
ω 𝑥 ≈ 𝑤 → 𝜑) |
93 | 2, 92 | sylbi 220 |
. 2
⊢ (𝑥 ∈ Fin → 𝜑) |
94 | 1, 93 | vtoclga 3478 |
1
⊢ (𝐴 ∈ Fin → 𝜏) |