Proof of Theorem pwfseqlem4
Step | Hyp | Ref
| Expression |
1 | | eqid 2622 |
. . . . . . . . . . 11
⊢ 𝑍 = 𝑍 |
2 | | eqid 2622 |
. . . . . . . . . . 11
⊢ (𝑊‘𝑍) = (𝑊‘𝑍) |
3 | 1, 2 | pm3.2i 471 |
. . . . . . . . . 10
⊢ (𝑍 = 𝑍 ∧ (𝑊‘𝑍) = (𝑊‘𝑍)) |
4 | | pwfseqlem4.w |
. . . . . . . . . . 11
⊢ 𝑊 = {〈𝑎, 𝑠〉 ∣ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑏 ∈ 𝑎 [(◡𝑠 “ {𝑏}) / 𝑣](𝑣𝐹(𝑠 ∩ (𝑣 × 𝑣))) = 𝑏))} |
5 | | pwfseqlem4.g |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐺:𝒫 𝐴–1-1→∪ 𝑛 ∈ ω (𝐴 ↑𝑚
𝑛)) |
6 | | omex 8540 |
. . . . . . . . . . . . . 14
⊢ ω
∈ V |
7 | | ovex 6678 |
. . . . . . . . . . . . . 14
⊢ (𝐴 ↑𝑚
𝑛) ∈
V |
8 | 6, 7 | iunex 7147 |
. . . . . . . . . . . . 13
⊢ ∪ 𝑛 ∈ ω (𝐴 ↑𝑚 𝑛) ∈ V |
9 | | f1dmex 7136 |
. . . . . . . . . . . . 13
⊢ ((𝐺:𝒫 𝐴–1-1→∪ 𝑛 ∈ ω (𝐴 ↑𝑚
𝑛) ∧ ∪ 𝑛 ∈ ω (𝐴 ↑𝑚 𝑛) ∈ V) → 𝒫
𝐴 ∈
V) |
10 | 5, 8, 9 | sylancl 694 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝒫 𝐴 ∈ V) |
11 | | pwexb 6975 |
. . . . . . . . . . . 12
⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V) |
12 | 10, 11 | sylibr 224 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐴 ∈ V) |
13 | | pwfseqlem4.x |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑋 ⊆ 𝐴) |
14 | | pwfseqlem4.h |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐻:ω–1-1-onto→𝑋) |
15 | | pwfseqlem4.ps |
. . . . . . . . . . . 12
⊢ (𝜓 ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ∧ ω ≼ 𝑥)) |
16 | | pwfseqlem4.k |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝜓) → 𝐾:∪ 𝑛 ∈ ω (𝑥 ↑𝑚
𝑛)–1-1→𝑥) |
17 | | pwfseqlem4.d |
. . . . . . . . . . . 12
⊢ 𝐷 = (𝐺‘{𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))}) |
18 | | pwfseqlem4.f |
. . . . . . . . . . . 12
⊢ 𝐹 = (𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷‘∩ {𝑧 ∈ ω ∣ ¬
(𝐷‘𝑧) ∈ 𝑥}))) |
19 | 5, 13, 14, 15, 16, 17, 18 | pwfseqlem4a 9483 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (𝑎𝐹𝑠) ∈ 𝐴) |
20 | | pwfseqlem4.z |
. . . . . . . . . . 11
⊢ 𝑍 = ∪
dom 𝑊 |
21 | 4, 12, 19, 20 | fpwwe2 9465 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑍𝑊(𝑊‘𝑍) ∧ (𝑍𝐹(𝑊‘𝑍)) ∈ 𝑍) ↔ (𝑍 = 𝑍 ∧ (𝑊‘𝑍) = (𝑊‘𝑍)))) |
22 | 3, 21 | mpbiri 248 |
. . . . . . . . 9
⊢ (𝜑 → (𝑍𝑊(𝑊‘𝑍) ∧ (𝑍𝐹(𝑊‘𝑍)) ∈ 𝑍)) |
23 | 22 | simprd 479 |
. . . . . . . 8
⊢ (𝜑 → (𝑍𝐹(𝑊‘𝑍)) ∈ 𝑍) |
24 | 22 | simpld 475 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑍𝑊(𝑊‘𝑍)) |
25 | 4, 12 | fpwwe2lem2 9454 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑍𝑊(𝑊‘𝑍) ↔ ((𝑍 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑍 × 𝑍)) ∧ ((𝑊‘𝑍) We 𝑍 ∧ ∀𝑏 ∈ 𝑍 [(◡(𝑊‘𝑍) “ {𝑏}) / 𝑣](𝑣𝐹((𝑊‘𝑍) ∩ (𝑣 × 𝑣))) = 𝑏)))) |
26 | 24, 25 | mpbid 222 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝑍 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑍 × 𝑍)) ∧ ((𝑊‘𝑍) We 𝑍 ∧ ∀𝑏 ∈ 𝑍 [(◡(𝑊‘𝑍) “ {𝑏}) / 𝑣](𝑣𝐹((𝑊‘𝑍) ∩ (𝑣 × 𝑣))) = 𝑏))) |
27 | 26 | simpld 475 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑍 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑍 × 𝑍))) |
28 | 27 | simpld 475 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑍 ⊆ 𝐴) |
29 | 12, 28 | ssexd 4805 |
. . . . . . . . 9
⊢ (𝜑 → 𝑍 ∈ V) |
30 | | sseq1 3626 |
. . . . . . . . . . . . . 14
⊢ (𝑎 = 𝑍 → (𝑎 ⊆ 𝐴 ↔ 𝑍 ⊆ 𝐴)) |
31 | | id 22 |
. . . . . . . . . . . . . . . 16
⊢ (𝑎 = 𝑍 → 𝑎 = 𝑍) |
32 | 31 | sqxpeqd 5141 |
. . . . . . . . . . . . . . 15
⊢ (𝑎 = 𝑍 → (𝑎 × 𝑎) = (𝑍 × 𝑍)) |
33 | 32 | sseq2d 3633 |
. . . . . . . . . . . . . 14
⊢ (𝑎 = 𝑍 → ((𝑊‘𝑍) ⊆ (𝑎 × 𝑎) ↔ (𝑊‘𝑍) ⊆ (𝑍 × 𝑍))) |
34 | | weeq2 5103 |
. . . . . . . . . . . . . 14
⊢ (𝑎 = 𝑍 → ((𝑊‘𝑍) We 𝑎 ↔ (𝑊‘𝑍) We 𝑍)) |
35 | 30, 33, 34 | 3anbi123d 1399 |
. . . . . . . . . . . . 13
⊢ (𝑎 = 𝑍 → ((𝑎 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑎 × 𝑎) ∧ (𝑊‘𝑍) We 𝑎) ↔ (𝑍 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑍 × 𝑍) ∧ (𝑊‘𝑍) We 𝑍))) |
36 | 35 | anbi2d 740 |
. . . . . . . . . . . 12
⊢ (𝑎 = 𝑍 → ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑎 × 𝑎) ∧ (𝑊‘𝑍) We 𝑎)) ↔ (𝜑 ∧ (𝑍 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑍 × 𝑍) ∧ (𝑊‘𝑍) We 𝑍)))) |
37 | | id 22 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑍 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑍 × 𝑍) ∧ (𝑊‘𝑍) We 𝑍) → (𝑍 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑍 × 𝑍) ∧ (𝑊‘𝑍) We 𝑍)) |
38 | 37 | 3expa 1265 |
. . . . . . . . . . . . . . 15
⊢ (((𝑍 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑍 × 𝑍)) ∧ (𝑊‘𝑍) We 𝑍) → (𝑍 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑍 × 𝑍) ∧ (𝑊‘𝑍) We 𝑍)) |
39 | 38 | adantrr 753 |
. . . . . . . . . . . . . 14
⊢ (((𝑍 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑍 × 𝑍)) ∧ ((𝑊‘𝑍) We 𝑍 ∧ ∀𝑏 ∈ 𝑍 [(◡(𝑊‘𝑍) “ {𝑏}) / 𝑣](𝑣𝐹((𝑊‘𝑍) ∩ (𝑣 × 𝑣))) = 𝑏)) → (𝑍 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑍 × 𝑍) ∧ (𝑊‘𝑍) We 𝑍)) |
40 | 26, 39 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑍 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑍 × 𝑍) ∧ (𝑊‘𝑍) We 𝑍)) |
41 | 40 | pm4.71i 664 |
. . . . . . . . . . . 12
⊢ (𝜑 ↔ (𝜑 ∧ (𝑍 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑍 × 𝑍) ∧ (𝑊‘𝑍) We 𝑍))) |
42 | 36, 41 | syl6bbr 278 |
. . . . . . . . . . 11
⊢ (𝑎 = 𝑍 → ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑎 × 𝑎) ∧ (𝑊‘𝑍) We 𝑎)) ↔ 𝜑)) |
43 | | oveq1 6657 |
. . . . . . . . . . . . 13
⊢ (𝑎 = 𝑍 → (𝑎𝐹(𝑊‘𝑍)) = (𝑍𝐹(𝑊‘𝑍))) |
44 | 43, 31 | eleq12d 2695 |
. . . . . . . . . . . 12
⊢ (𝑎 = 𝑍 → ((𝑎𝐹(𝑊‘𝑍)) ∈ 𝑎 ↔ (𝑍𝐹(𝑊‘𝑍)) ∈ 𝑍)) |
45 | | breq1 4656 |
. . . . . . . . . . . 12
⊢ (𝑎 = 𝑍 → (𝑎 ≺ ω ↔ 𝑍 ≺ ω)) |
46 | 44, 45 | imbi12d 334 |
. . . . . . . . . . 11
⊢ (𝑎 = 𝑍 → (((𝑎𝐹(𝑊‘𝑍)) ∈ 𝑎 → 𝑎 ≺ ω) ↔ ((𝑍𝐹(𝑊‘𝑍)) ∈ 𝑍 → 𝑍 ≺ ω))) |
47 | 42, 46 | imbi12d 334 |
. . . . . . . . . 10
⊢ (𝑎 = 𝑍 → (((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑎 × 𝑎) ∧ (𝑊‘𝑍) We 𝑎)) → ((𝑎𝐹(𝑊‘𝑍)) ∈ 𝑎 → 𝑎 ≺ ω)) ↔ (𝜑 → ((𝑍𝐹(𝑊‘𝑍)) ∈ 𝑍 → 𝑍 ≺ ω)))) |
48 | | fvex 6201 |
. . . . . . . . . . 11
⊢ (𝑊‘𝑍) ∈ V |
49 | | sseq1 3626 |
. . . . . . . . . . . . . 14
⊢ (𝑠 = (𝑊‘𝑍) → (𝑠 ⊆ (𝑎 × 𝑎) ↔ (𝑊‘𝑍) ⊆ (𝑎 × 𝑎))) |
50 | | weeq1 5102 |
. . . . . . . . . . . . . 14
⊢ (𝑠 = (𝑊‘𝑍) → (𝑠 We 𝑎 ↔ (𝑊‘𝑍) We 𝑎)) |
51 | 49, 50 | 3anbi23d 1402 |
. . . . . . . . . . . . 13
⊢ (𝑠 = (𝑊‘𝑍) → ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ↔ (𝑎 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑎 × 𝑎) ∧ (𝑊‘𝑍) We 𝑎))) |
52 | 51 | anbi2d 740 |
. . . . . . . . . . . 12
⊢ (𝑠 = (𝑊‘𝑍) → ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) ↔ (𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑎 × 𝑎) ∧ (𝑊‘𝑍) We 𝑎)))) |
53 | | oveq2 6658 |
. . . . . . . . . . . . . 14
⊢ (𝑠 = (𝑊‘𝑍) → (𝑎𝐹𝑠) = (𝑎𝐹(𝑊‘𝑍))) |
54 | 53 | eleq1d 2686 |
. . . . . . . . . . . . 13
⊢ (𝑠 = (𝑊‘𝑍) → ((𝑎𝐹𝑠) ∈ 𝑎 ↔ (𝑎𝐹(𝑊‘𝑍)) ∈ 𝑎)) |
55 | 54 | imbi1d 331 |
. . . . . . . . . . . 12
⊢ (𝑠 = (𝑊‘𝑍) → (((𝑎𝐹𝑠) ∈ 𝑎 → 𝑎 ≺ ω) ↔ ((𝑎𝐹(𝑊‘𝑍)) ∈ 𝑎 → 𝑎 ≺ ω))) |
56 | 52, 55 | imbi12d 334 |
. . . . . . . . . . 11
⊢ (𝑠 = (𝑊‘𝑍) → (((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → ((𝑎𝐹𝑠) ∈ 𝑎 → 𝑎 ≺ ω)) ↔ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑎 × 𝑎) ∧ (𝑊‘𝑍) We 𝑎)) → ((𝑎𝐹(𝑊‘𝑍)) ∈ 𝑎 → 𝑎 ≺ ω)))) |
57 | | omelon 8543 |
. . . . . . . . . . . . . . 15
⊢ ω
∈ On |
58 | | onenon 8775 |
. . . . . . . . . . . . . . 15
⊢ (ω
∈ On → ω ∈ dom card) |
59 | 57, 58 | ax-mp 5 |
. . . . . . . . . . . . . 14
⊢ ω
∈ dom card |
60 | | simpr3 1069 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → 𝑠 We 𝑎) |
61 | | 19.8a 2052 |
. . . . . . . . . . . . . . . 16
⊢ (𝑠 We 𝑎 → ∃𝑠 𝑠 We 𝑎) |
62 | 60, 61 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → ∃𝑠 𝑠 We 𝑎) |
63 | | ween 8858 |
. . . . . . . . . . . . . . 15
⊢ (𝑎 ∈ dom card ↔
∃𝑠 𝑠 We 𝑎) |
64 | 62, 63 | sylibr 224 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → 𝑎 ∈ dom card) |
65 | | domtri2 8815 |
. . . . . . . . . . . . . 14
⊢ ((ω
∈ dom card ∧ 𝑎
∈ dom card) → (ω ≼ 𝑎 ↔ ¬ 𝑎 ≺ ω)) |
66 | 59, 64, 65 | sylancr 695 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (ω ≼ 𝑎 ↔ ¬ 𝑎 ≺ ω)) |
67 | | nfv 1843 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑟(𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)) |
68 | | nfcv 2764 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑟𝑎 |
69 | | nfmpt22 6723 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑟(𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷‘∩ {𝑧 ∈ ω ∣ ¬
(𝐷‘𝑧) ∈ 𝑥}))) |
70 | 18, 69 | nfcxfr 2762 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑟𝐹 |
71 | | nfcv 2764 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑟𝑠 |
72 | 68, 70, 71 | nfov 6676 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑟(𝑎𝐹𝑠) |
73 | 72 | nfel1 2779 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑟(𝑎𝐹𝑠) ∈ (𝐴 ∖ 𝑎) |
74 | 67, 73 | nfim 1825 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑟((𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑠) ∈ (𝐴 ∖ 𝑎)) |
75 | | sseq1 3626 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑟 = 𝑠 → (𝑟 ⊆ (𝑎 × 𝑎) ↔ 𝑠 ⊆ (𝑎 × 𝑎))) |
76 | | weeq1 5102 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑟 = 𝑠 → (𝑟 We 𝑎 ↔ 𝑠 We 𝑎)) |
77 | 75, 76 | 3anbi23d 1402 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑟 = 𝑠 → ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ↔ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎))) |
78 | 77 | anbi1d 741 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑟 = 𝑠 → (((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎) ↔ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎))) |
79 | 78 | anbi2d 740 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑟 = 𝑠 → ((𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) ↔ (𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)))) |
80 | | oveq2 6658 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑟 = 𝑠 → (𝑎𝐹𝑟) = (𝑎𝐹𝑠)) |
81 | 80 | eleq1d 2686 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑟 = 𝑠 → ((𝑎𝐹𝑟) ∈ (𝐴 ∖ 𝑎) ↔ (𝑎𝐹𝑠) ∈ (𝐴 ∖ 𝑎))) |
82 | 79, 81 | imbi12d 334 |
. . . . . . . . . . . . . . . 16
⊢ (𝑟 = 𝑠 → (((𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑟) ∈ (𝐴 ∖ 𝑎)) ↔ ((𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑠) ∈ (𝐴 ∖ 𝑎)))) |
83 | | nfv 1843 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑥(𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) |
84 | | nfcv 2764 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑥𝑎 |
85 | | nfmpt21 6722 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
Ⅎ𝑥(𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷‘∩ {𝑧 ∈ ω ∣ ¬
(𝐷‘𝑧) ∈ 𝑥}))) |
86 | 18, 85 | nfcxfr 2762 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑥𝐹 |
87 | | nfcv 2764 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑥𝑟 |
88 | 84, 86, 87 | nfov 6676 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑥(𝑎𝐹𝑟) |
89 | 88 | nfel1 2779 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑥(𝑎𝐹𝑟) ∈ (𝐴 ∖ 𝑎) |
90 | 83, 89 | nfim 1825 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑥((𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑟) ∈ (𝐴 ∖ 𝑎)) |
91 | | sseq1 3626 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 = 𝑎 → (𝑥 ⊆ 𝐴 ↔ 𝑎 ⊆ 𝐴)) |
92 | | xpeq12 5134 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑥 = 𝑎 ∧ 𝑥 = 𝑎) → (𝑥 × 𝑥) = (𝑎 × 𝑎)) |
93 | 92 | anidms 677 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑥 = 𝑎 → (𝑥 × 𝑥) = (𝑎 × 𝑎)) |
94 | 93 | sseq2d 3633 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 = 𝑎 → (𝑟 ⊆ (𝑥 × 𝑥) ↔ 𝑟 ⊆ (𝑎 × 𝑎))) |
95 | | weeq2 5103 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 = 𝑎 → (𝑟 We 𝑥 ↔ 𝑟 We 𝑎)) |
96 | 91, 94, 95 | 3anbi123d 1399 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 = 𝑎 → ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ↔ (𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎))) |
97 | | breq2 4657 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 = 𝑎 → (ω ≼ 𝑥 ↔ ω ≼ 𝑎)) |
98 | 96, 97 | anbi12d 747 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 = 𝑎 → (((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ∧ ω ≼ 𝑥) ↔ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎))) |
99 | 15, 98 | syl5bb 272 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 = 𝑎 → (𝜓 ↔ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎))) |
100 | 99 | anbi2d 740 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = 𝑎 → ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)))) |
101 | | oveq1 6657 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 = 𝑎 → (𝑥𝐹𝑟) = (𝑎𝐹𝑟)) |
102 | | difeq2 3722 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 = 𝑎 → (𝐴 ∖ 𝑥) = (𝐴 ∖ 𝑎)) |
103 | 101, 102 | eleq12d 2695 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = 𝑎 → ((𝑥𝐹𝑟) ∈ (𝐴 ∖ 𝑥) ↔ (𝑎𝐹𝑟) ∈ (𝐴 ∖ 𝑎))) |
104 | 100, 103 | imbi12d 334 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 = 𝑎 → (((𝜑 ∧ 𝜓) → (𝑥𝐹𝑟) ∈ (𝐴 ∖ 𝑥)) ↔ ((𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑟) ∈ (𝐴 ∖ 𝑎)))) |
105 | 5, 13, 14, 15, 16, 17, 18 | pwfseqlem3 9482 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝜓) → (𝑥𝐹𝑟) ∈ (𝐴 ∖ 𝑥)) |
106 | 90, 104, 105 | chvar 2262 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑎 × 𝑎) ∧ 𝑟 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑟) ∈ (𝐴 ∖ 𝑎)) |
107 | 74, 82, 106 | chvar 2262 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)) → (𝑎𝐹𝑠) ∈ (𝐴 ∖ 𝑎)) |
108 | 107 | eldifbd 3587 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎) ∧ ω ≼ 𝑎)) → ¬ (𝑎𝐹𝑠) ∈ 𝑎) |
109 | 108 | expr 643 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (ω ≼ 𝑎 → ¬ (𝑎𝐹𝑠) ∈ 𝑎)) |
110 | 66, 109 | sylbird 250 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (¬ 𝑎 ≺ ω → ¬ (𝑎𝐹𝑠) ∈ 𝑎)) |
111 | 110 | con4d 114 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → ((𝑎𝐹𝑠) ∈ 𝑎 → 𝑎 ≺ ω)) |
112 | 48, 56, 111 | vtocl 3259 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ (𝑊‘𝑍) ⊆ (𝑎 × 𝑎) ∧ (𝑊‘𝑍) We 𝑎)) → ((𝑎𝐹(𝑊‘𝑍)) ∈ 𝑎 → 𝑎 ≺ ω)) |
113 | 47, 112 | vtoclg 3266 |
. . . . . . . . 9
⊢ (𝑍 ∈ V → (𝜑 → ((𝑍𝐹(𝑊‘𝑍)) ∈ 𝑍 → 𝑍 ≺ ω))) |
114 | 29, 113 | mpcom 38 |
. . . . . . . 8
⊢ (𝜑 → ((𝑍𝐹(𝑊‘𝑍)) ∈ 𝑍 → 𝑍 ≺ ω)) |
115 | 23, 114 | mpd 15 |
. . . . . . 7
⊢ (𝜑 → 𝑍 ≺ ω) |
116 | | isfinite 8549 |
. . . . . . 7
⊢ (𝑍 ∈ Fin ↔ 𝑍 ≺
ω) |
117 | 115, 116 | sylibr 224 |
. . . . . 6
⊢ (𝜑 → 𝑍 ∈ Fin) |
118 | 5, 13, 14, 15, 16, 17, 18 | pwfseqlem2 9481 |
. . . . . 6
⊢ ((𝑍 ∈ Fin ∧ (𝑊‘𝑍) ∈ V) → (𝑍𝐹(𝑊‘𝑍)) = (𝐻‘(card‘𝑍))) |
119 | 117, 48, 118 | sylancl 694 |
. . . . 5
⊢ (𝜑 → (𝑍𝐹(𝑊‘𝑍)) = (𝐻‘(card‘𝑍))) |
120 | 119, 23 | eqeltrrd 2702 |
. . . 4
⊢ (𝜑 → (𝐻‘(card‘𝑍)) ∈ 𝑍) |
121 | 4, 12, 24 | fpwwe2lem3 9455 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝐻‘(card‘𝑍)) ∈ 𝑍) → ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))})𝐹((𝑊‘𝑍) ∩ ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) × (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))})))) = (𝐻‘(card‘𝑍))) |
122 | 120, 121 | mpdan 702 |
. . . . . . . . 9
⊢ (𝜑 → ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))})𝐹((𝑊‘𝑍) ∩ ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) × (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))})))) = (𝐻‘(card‘𝑍))) |
123 | | cnvimass 5485 |
. . . . . . . . . . . 12
⊢ (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ⊆ dom (𝑊‘𝑍) |
124 | 27 | simprd 479 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑊‘𝑍) ⊆ (𝑍 × 𝑍)) |
125 | | dmss 5323 |
. . . . . . . . . . . . . 14
⊢ ((𝑊‘𝑍) ⊆ (𝑍 × 𝑍) → dom (𝑊‘𝑍) ⊆ dom (𝑍 × 𝑍)) |
126 | 124, 125 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → dom (𝑊‘𝑍) ⊆ dom (𝑍 × 𝑍)) |
127 | | dmxpss 5565 |
. . . . . . . . . . . . 13
⊢ dom
(𝑍 × 𝑍) ⊆ 𝑍 |
128 | 126, 127 | syl6ss 3615 |
. . . . . . . . . . . 12
⊢ (𝜑 → dom (𝑊‘𝑍) ⊆ 𝑍) |
129 | 123, 128 | syl5ss 3614 |
. . . . . . . . . . 11
⊢ (𝜑 → (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ⊆ 𝑍) |
130 | 117, 129 | ssfid 8183 |
. . . . . . . . . 10
⊢ (𝜑 → (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ∈ Fin) |
131 | 48 | inex1 4799 |
. . . . . . . . . 10
⊢ ((𝑊‘𝑍) ∩ ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) × (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}))) ∈ V |
132 | 5, 13, 14, 15, 16, 17, 18 | pwfseqlem2 9481 |
. . . . . . . . . 10
⊢ (((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ∈ Fin ∧ ((𝑊‘𝑍) ∩ ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) × (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}))) ∈ V) → ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))})𝐹((𝑊‘𝑍) ∩ ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) × (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))})))) = (𝐻‘(card‘(◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))})))) |
133 | 130, 131,
132 | sylancl 694 |
. . . . . . . . 9
⊢ (𝜑 → ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))})𝐹((𝑊‘𝑍) ∩ ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) × (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))})))) = (𝐻‘(card‘(◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))})))) |
134 | 122, 133 | eqtr3d 2658 |
. . . . . . . 8
⊢ (𝜑 → (𝐻‘(card‘𝑍)) = (𝐻‘(card‘(◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))})))) |
135 | | f1of1 6136 |
. . . . . . . . . 10
⊢ (𝐻:ω–1-1-onto→𝑋 → 𝐻:ω–1-1→𝑋) |
136 | 14, 135 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐻:ω–1-1→𝑋) |
137 | | ficardom 8787 |
. . . . . . . . . 10
⊢ (𝑍 ∈ Fin →
(card‘𝑍) ∈
ω) |
138 | 117, 137 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (card‘𝑍) ∈
ω) |
139 | | ficardom 8787 |
. . . . . . . . . 10
⊢ ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ∈ Fin → (card‘(◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))})) ∈ ω) |
140 | 130, 139 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (card‘(◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))})) ∈ ω) |
141 | | f1fveq 6519 |
. . . . . . . . 9
⊢ ((𝐻:ω–1-1→𝑋 ∧ ((card‘𝑍) ∈ ω ∧ (card‘(◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))})) ∈ ω)) → ((𝐻‘(card‘𝑍)) = (𝐻‘(card‘(◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}))) ↔ (card‘𝑍) = (card‘(◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))})))) |
142 | 136, 138,
140, 141 | syl12anc 1324 |
. . . . . . . 8
⊢ (𝜑 → ((𝐻‘(card‘𝑍)) = (𝐻‘(card‘(◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}))) ↔ (card‘𝑍) = (card‘(◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))})))) |
143 | 134, 142 | mpbid 222 |
. . . . . . 7
⊢ (𝜑 → (card‘𝑍) = (card‘(◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}))) |
144 | 143 | eqcomd 2628 |
. . . . . 6
⊢ (𝜑 → (card‘(◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))})) = (card‘𝑍)) |
145 | | finnum 8774 |
. . . . . . . 8
⊢ ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ∈ Fin → (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ∈ dom card) |
146 | 130, 145 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ∈ dom card) |
147 | | finnum 8774 |
. . . . . . . 8
⊢ (𝑍 ∈ Fin → 𝑍 ∈ dom
card) |
148 | 117, 147 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝑍 ∈ dom card) |
149 | | carden2 8813 |
. . . . . . 7
⊢ (((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ∈ dom card ∧ 𝑍 ∈ dom card) → ((card‘(◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))})) = (card‘𝑍) ↔ (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ≈ 𝑍)) |
150 | 146, 148,
149 | syl2anc 693 |
. . . . . 6
⊢ (𝜑 → ((card‘(◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))})) = (card‘𝑍) ↔ (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ≈ 𝑍)) |
151 | 144, 150 | mpbid 222 |
. . . . 5
⊢ (𝜑 → (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ≈ 𝑍) |
152 | | dfpss2 3692 |
. . . . . . . 8
⊢ ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ⊊ 𝑍 ↔ ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ⊆ 𝑍 ∧ ¬ (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) = 𝑍)) |
153 | 152 | baib 944 |
. . . . . . 7
⊢ ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ⊆ 𝑍 → ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ⊊ 𝑍 ↔ ¬ (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) = 𝑍)) |
154 | 129, 153 | syl 17 |
. . . . . 6
⊢ (𝜑 → ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ⊊ 𝑍 ↔ ¬ (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) = 𝑍)) |
155 | | php3 8146 |
. . . . . . . . 9
⊢ ((𝑍 ∈ Fin ∧ (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ⊊ 𝑍) → (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ≺ 𝑍) |
156 | | sdomnen 7984 |
. . . . . . . . 9
⊢ ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ≺ 𝑍 → ¬ (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ≈ 𝑍) |
157 | 155, 156 | syl 17 |
. . . . . . . 8
⊢ ((𝑍 ∈ Fin ∧ (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ⊊ 𝑍) → ¬ (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ≈ 𝑍) |
158 | 157 | ex 450 |
. . . . . . 7
⊢ (𝑍 ∈ Fin → ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ⊊ 𝑍 → ¬ (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ≈ 𝑍)) |
159 | 117, 158 | syl 17 |
. . . . . 6
⊢ (𝜑 → ((◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ⊊ 𝑍 → ¬ (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ≈ 𝑍)) |
160 | 154, 159 | sylbird 250 |
. . . . 5
⊢ (𝜑 → (¬ (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) = 𝑍 → ¬ (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ≈ 𝑍)) |
161 | 151, 160 | mt4d 152 |
. . . 4
⊢ (𝜑 → (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) = 𝑍) |
162 | 120, 161 | eleqtrrd 2704 |
. . 3
⊢ (𝜑 → (𝐻‘(card‘𝑍)) ∈ (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))})) |
163 | | fvex 6201 |
. . . 4
⊢ (𝐻‘(card‘𝑍)) ∈ V |
164 | 163 | eliniseg 5494 |
. . . 4
⊢ ((𝐻‘(card‘𝑍)) ∈ V → ((𝐻‘(card‘𝑍)) ∈ (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ↔ (𝐻‘(card‘𝑍))(𝑊‘𝑍)(𝐻‘(card‘𝑍)))) |
165 | 163, 164 | ax-mp 5 |
. . 3
⊢ ((𝐻‘(card‘𝑍)) ∈ (◡(𝑊‘𝑍) “ {(𝐻‘(card‘𝑍))}) ↔ (𝐻‘(card‘𝑍))(𝑊‘𝑍)(𝐻‘(card‘𝑍))) |
166 | 162, 165 | sylib 208 |
. 2
⊢ (𝜑 → (𝐻‘(card‘𝑍))(𝑊‘𝑍)(𝐻‘(card‘𝑍))) |
167 | 26 | simprd 479 |
. . . . 5
⊢ (𝜑 → ((𝑊‘𝑍) We 𝑍 ∧ ∀𝑏 ∈ 𝑍 [(◡(𝑊‘𝑍) “ {𝑏}) / 𝑣](𝑣𝐹((𝑊‘𝑍) ∩ (𝑣 × 𝑣))) = 𝑏)) |
168 | 167 | simpld 475 |
. . . 4
⊢ (𝜑 → (𝑊‘𝑍) We 𝑍) |
169 | | weso 5105 |
. . . 4
⊢ ((𝑊‘𝑍) We 𝑍 → (𝑊‘𝑍) Or 𝑍) |
170 | 168, 169 | syl 17 |
. . 3
⊢ (𝜑 → (𝑊‘𝑍) Or 𝑍) |
171 | | sonr 5056 |
. . 3
⊢ (((𝑊‘𝑍) Or 𝑍 ∧ (𝐻‘(card‘𝑍)) ∈ 𝑍) → ¬ (𝐻‘(card‘𝑍))(𝑊‘𝑍)(𝐻‘(card‘𝑍))) |
172 | 170, 120,
171 | syl2anc 693 |
. 2
⊢ (𝜑 → ¬ (𝐻‘(card‘𝑍))(𝑊‘𝑍)(𝐻‘(card‘𝑍))) |
173 | 166, 172 | pm2.65i 185 |
1
⊢ ¬
𝜑 |