| Step | Hyp | Ref
| Expression |
| 1 | | findcard2.4 |
. 2
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) |
| 2 | | isfi 9016 |
. . 3
⊢ (𝑥 ∈ Fin ↔ ∃𝑤 ∈ ω 𝑥 ≈ 𝑤) |
| 3 | | breq2 5147 |
. . . . . . . 8
⊢ (𝑤 = ∅ → (𝑥 ≈ 𝑤 ↔ 𝑥 ≈ ∅)) |
| 4 | 3 | imbi1d 341 |
. . . . . . 7
⊢ (𝑤 = ∅ → ((𝑥 ≈ 𝑤 → 𝜑) ↔ (𝑥 ≈ ∅ → 𝜑))) |
| 5 | 4 | albidv 1920 |
. . . . . 6
⊢ (𝑤 = ∅ → (∀𝑥(𝑥 ≈ 𝑤 → 𝜑) ↔ ∀𝑥(𝑥 ≈ ∅ → 𝜑))) |
| 6 | | breq2 5147 |
. . . . . . . 8
⊢ (𝑤 = 𝑣 → (𝑥 ≈ 𝑤 ↔ 𝑥 ≈ 𝑣)) |
| 7 | 6 | imbi1d 341 |
. . . . . . 7
⊢ (𝑤 = 𝑣 → ((𝑥 ≈ 𝑤 → 𝜑) ↔ (𝑥 ≈ 𝑣 → 𝜑))) |
| 8 | 7 | albidv 1920 |
. . . . . 6
⊢ (𝑤 = 𝑣 → (∀𝑥(𝑥 ≈ 𝑤 → 𝜑) ↔ ∀𝑥(𝑥 ≈ 𝑣 → 𝜑))) |
| 9 | | breq2 5147 |
. . . . . . . 8
⊢ (𝑤 = suc 𝑣 → (𝑥 ≈ 𝑤 ↔ 𝑥 ≈ suc 𝑣)) |
| 10 | 9 | imbi1d 341 |
. . . . . . 7
⊢ (𝑤 = suc 𝑣 → ((𝑥 ≈ 𝑤 → 𝜑) ↔ (𝑥 ≈ suc 𝑣 → 𝜑))) |
| 11 | 10 | albidv 1920 |
. . . . . 6
⊢ (𝑤 = suc 𝑣 → (∀𝑥(𝑥 ≈ 𝑤 → 𝜑) ↔ ∀𝑥(𝑥 ≈ suc 𝑣 → 𝜑))) |
| 12 | | en0 9058 |
. . . . . . . 8
⊢ (𝑥 ≈ ∅ ↔ 𝑥 = ∅) |
| 13 | | findcard2.5 |
. . . . . . . . 9
⊢ 𝜓 |
| 14 | | findcard2.1 |
. . . . . . . . 9
⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓)) |
| 15 | 13, 14 | mpbiri 258 |
. . . . . . . 8
⊢ (𝑥 = ∅ → 𝜑) |
| 16 | 12, 15 | sylbi 217 |
. . . . . . 7
⊢ (𝑥 ≈ ∅ → 𝜑) |
| 17 | 16 | ax-gen 1795 |
. . . . . 6
⊢
∀𝑥(𝑥 ≈ ∅ → 𝜑) |
| 18 | | nnon 7893 |
. . . . . . . . . . . 12
⊢ (𝑣 ∈ ω → 𝑣 ∈ On) |
| 19 | | rexdif1en 9198 |
. . . . . . . . . . . 12
⊢ ((𝑣 ∈ On ∧ 𝑤 ≈ suc 𝑣) → ∃𝑧 ∈ 𝑤 (𝑤 ∖ {𝑧}) ≈ 𝑣) |
| 20 | 18, 19 | sylan 580 |
. . . . . . . . . . 11
⊢ ((𝑣 ∈ ω ∧ 𝑤 ≈ suc 𝑣) → ∃𝑧 ∈ 𝑤 (𝑤 ∖ {𝑧}) ≈ 𝑣) |
| 21 | | snssi 4808 |
. . . . . . . . . . . . . . . 16
⊢ (𝑧 ∈ 𝑤 → {𝑧} ⊆ 𝑤) |
| 22 | | uncom 4158 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑤 ∖ {𝑧}) ∪ {𝑧}) = ({𝑧} ∪ (𝑤 ∖ {𝑧})) |
| 23 | | undif 4482 |
. . . . . . . . . . . . . . . . . 18
⊢ ({𝑧} ⊆ 𝑤 ↔ ({𝑧} ∪ (𝑤 ∖ {𝑧})) = 𝑤) |
| 24 | 23 | biimpi 216 |
. . . . . . . . . . . . . . . . 17
⊢ ({𝑧} ⊆ 𝑤 → ({𝑧} ∪ (𝑤 ∖ {𝑧})) = 𝑤) |
| 25 | 22, 24 | eqtrid 2789 |
. . . . . . . . . . . . . . . 16
⊢ ({𝑧} ⊆ 𝑤 → ((𝑤 ∖ {𝑧}) ∪ {𝑧}) = 𝑤) |
| 26 | | vex 3484 |
. . . . . . . . . . . . . . . . . . 19
⊢ 𝑤 ∈ V |
| 27 | 26 | difexi 5330 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑤 ∖ {𝑧}) ∈ V |
| 28 | | breq1 5146 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 = (𝑤 ∖ {𝑧}) → (𝑦 ≈ 𝑣 ↔ (𝑤 ∖ {𝑧}) ≈ 𝑣)) |
| 29 | 28 | anbi2d 630 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 = (𝑤 ∖ {𝑧}) → ((𝑣 ∈ ω ∧ 𝑦 ≈ 𝑣) ↔ (𝑣 ∈ ω ∧ (𝑤 ∖ {𝑧}) ≈ 𝑣))) |
| 30 | | uneq1 4161 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 = (𝑤 ∖ {𝑧}) → (𝑦 ∪ {𝑧}) = ((𝑤 ∖ {𝑧}) ∪ {𝑧})) |
| 31 | 30 | sbceq1d 3793 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 = (𝑤 ∖ {𝑧}) → ([(𝑦 ∪ {𝑧}) / 𝑥]𝜑 ↔ [((𝑤 ∖ {𝑧}) ∪ {𝑧}) / 𝑥]𝜑)) |
| 32 | 31 | imbi2d 340 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 = (𝑤 ∖ {𝑧}) → ((∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [(𝑦 ∪ {𝑧}) / 𝑥]𝜑) ↔ (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [((𝑤 ∖ {𝑧}) ∪ {𝑧}) / 𝑥]𝜑))) |
| 33 | 29, 32 | imbi12d 344 |
. . . . . . . . . . . . . . . . . 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)))) |
| 34 | | breq1 5146 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 = 𝑦 → (𝑥 ≈ 𝑣 ↔ 𝑦 ≈ 𝑣)) |
| 35 | | findcard2.2 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) |
| 36 | 34, 35 | imbi12d 344 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 = 𝑦 → ((𝑥 ≈ 𝑣 → 𝜑) ↔ (𝑦 ≈ 𝑣 → 𝜒))) |
| 37 | 36 | spvv 1996 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → (𝑦 ≈ 𝑣 → 𝜒)) |
| 38 | | rspe 3249 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑣 ∈ ω ∧ 𝑦 ≈ 𝑣) → ∃𝑣 ∈ ω 𝑦 ≈ 𝑣) |
| 39 | | isfi 9016 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 ∈ Fin ↔ ∃𝑣 ∈ ω 𝑦 ≈ 𝑣) |
| 40 | 38, 39 | sylibr 234 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑣 ∈ ω ∧ 𝑦 ≈ 𝑣) → 𝑦 ∈ Fin) |
| 41 | | pm2.27 42 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 ≈ 𝑣 → ((𝑦 ≈ 𝑣 → 𝜒) → 𝜒)) |
| 42 | 41 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑣 ∈ ω ∧ 𝑦 ≈ 𝑣) → ((𝑦 ≈ 𝑣 → 𝜒) → 𝜒)) |
| 43 | | findcard2.6 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 ∈ Fin → (𝜒 → 𝜃)) |
| 44 | 40, 42, 43 | sylsyld 61 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑣 ∈ ω ∧ 𝑦 ≈ 𝑣) → ((𝑦 ≈ 𝑣 → 𝜒) → 𝜃)) |
| 45 | 37, 44 | syl5 34 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑣 ∈ ω ∧ 𝑦 ≈ 𝑣) → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → 𝜃)) |
| 46 | | vex 3484 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 𝑦 ∈ V |
| 47 | | vsnex 5434 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ {𝑧} ∈ V |
| 48 | 46, 47 | unex 7764 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 ∪ {𝑧}) ∈ V |
| 49 | | findcard2.3 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝜑 ↔ 𝜃)) |
| 50 | 48, 49 | sbcie 3830 |
. . . . . . . . . . . . . . . . . . 19
⊢
([(𝑦 ∪
{𝑧}) / 𝑥]𝜑 ↔ 𝜃) |
| 51 | 45, 50 | imbitrrdi 252 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑣 ∈ ω ∧ 𝑦 ≈ 𝑣) → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [(𝑦 ∪ {𝑧}) / 𝑥]𝜑)) |
| 52 | 27, 33, 51 | vtocl 3558 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑣 ∈ ω ∧ (𝑤 ∖ {𝑧}) ≈ 𝑣) → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [((𝑤 ∖ {𝑧}) ∪ {𝑧}) / 𝑥]𝜑)) |
| 53 | | dfsbcq 3790 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑤 ∖ {𝑧}) ∪ {𝑧}) = 𝑤 → ([((𝑤 ∖ {𝑧}) ∪ {𝑧}) / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝜑)) |
| 54 | 53 | imbi2d 340 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑤 ∖ {𝑧}) ∪ {𝑧}) = 𝑤 → ((∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [((𝑤 ∖ {𝑧}) ∪ {𝑧}) / 𝑥]𝜑) ↔ (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑))) |
| 55 | 52, 54 | imbitrid 244 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑤 ∖ {𝑧}) ∪ {𝑧}) = 𝑤 → ((𝑣 ∈ ω ∧ (𝑤 ∖ {𝑧}) ≈ 𝑣) → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑))) |
| 56 | 21, 25, 55 | 3syl 18 |
. . . . . . . . . . . . . . 15
⊢ (𝑧 ∈ 𝑤 → ((𝑣 ∈ ω ∧ (𝑤 ∖ {𝑧}) ≈ 𝑣) → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑))) |
| 57 | 56 | expd 415 |
. . . . . . . . . . . . . 14
⊢ (𝑧 ∈ 𝑤 → (𝑣 ∈ ω → ((𝑤 ∖ {𝑧}) ≈ 𝑣 → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑)))) |
| 58 | 57 | com12 32 |
. . . . . . . . . . . . 13
⊢ (𝑣 ∈ ω → (𝑧 ∈ 𝑤 → ((𝑤 ∖ {𝑧}) ≈ 𝑣 → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑)))) |
| 59 | 58 | rexlimdv 3153 |
. . . . . . . . . . . 12
⊢ (𝑣 ∈ ω →
(∃𝑧 ∈ 𝑤 (𝑤 ∖ {𝑧}) ≈ 𝑣 → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑))) |
| 60 | 59 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝑣 ∈ ω ∧ 𝑤 ≈ suc 𝑣) → (∃𝑧 ∈ 𝑤 (𝑤 ∖ {𝑧}) ≈ 𝑣 → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑))) |
| 61 | 20, 60 | mpd 15 |
. . . . . . . . . 10
⊢ ((𝑣 ∈ ω ∧ 𝑤 ≈ suc 𝑣) → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑)) |
| 62 | 61 | ex 412 |
. . . . . . . . 9
⊢ (𝑣 ∈ ω → (𝑤 ≈ suc 𝑣 → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑))) |
| 63 | 62 | com23 86 |
. . . . . . . 8
⊢ (𝑣 ∈ ω →
(∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → (𝑤 ≈ suc 𝑣 → [𝑤 / 𝑥]𝜑))) |
| 64 | 63 | alrimdv 1929 |
. . . . . . 7
⊢ (𝑣 ∈ ω →
(∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → ∀𝑤(𝑤 ≈ suc 𝑣 → [𝑤 / 𝑥]𝜑))) |
| 65 | | nfv 1914 |
. . . . . . . 8
⊢
Ⅎ𝑤(𝑥 ≈ suc 𝑣 → 𝜑) |
| 66 | | nfv 1914 |
. . . . . . . . 9
⊢
Ⅎ𝑥 𝑤 ≈ suc 𝑣 |
| 67 | | nfsbc1v 3808 |
. . . . . . . . 9
⊢
Ⅎ𝑥[𝑤 / 𝑥]𝜑 |
| 68 | 66, 67 | nfim 1896 |
. . . . . . . 8
⊢
Ⅎ𝑥(𝑤 ≈ suc 𝑣 → [𝑤 / 𝑥]𝜑) |
| 69 | | breq1 5146 |
. . . . . . . . 9
⊢ (𝑥 = 𝑤 → (𝑥 ≈ suc 𝑣 ↔ 𝑤 ≈ suc 𝑣)) |
| 70 | | sbceq1a 3799 |
. . . . . . . . 9
⊢ (𝑥 = 𝑤 → (𝜑 ↔ [𝑤 / 𝑥]𝜑)) |
| 71 | 69, 70 | imbi12d 344 |
. . . . . . . 8
⊢ (𝑥 = 𝑤 → ((𝑥 ≈ suc 𝑣 → 𝜑) ↔ (𝑤 ≈ suc 𝑣 → [𝑤 / 𝑥]𝜑))) |
| 72 | 65, 68, 71 | cbvalv1 2343 |
. . . . . . 7
⊢
(∀𝑥(𝑥 ≈ suc 𝑣 → 𝜑) ↔ ∀𝑤(𝑤 ≈ suc 𝑣 → [𝑤 / 𝑥]𝜑)) |
| 73 | 64, 72 | imbitrrdi 252 |
. . . . . 6
⊢ (𝑣 ∈ ω →
(∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → ∀𝑥(𝑥 ≈ suc 𝑣 → 𝜑))) |
| 74 | 5, 8, 11, 17, 73 | finds1 7921 |
. . . . 5
⊢ (𝑤 ∈ ω →
∀𝑥(𝑥 ≈ 𝑤 → 𝜑)) |
| 75 | 74 | 19.21bi 2189 |
. . . 4
⊢ (𝑤 ∈ ω → (𝑥 ≈ 𝑤 → 𝜑)) |
| 76 | 75 | rexlimiv 3148 |
. . 3
⊢
(∃𝑤 ∈
ω 𝑥 ≈ 𝑤 → 𝜑) |
| 77 | 2, 76 | sylbi 217 |
. 2
⊢ (𝑥 ∈ Fin → 𝜑) |
| 78 | 1, 77 | vtoclga 3577 |
1
⊢ (𝐴 ∈ Fin → 𝜏) |