Step | Hyp | Ref
| Expression |
1 | | rankon 8541 |
. . 3
⊢
(rank‘𝐴)
∈ On |
2 | | onzsl 6938 |
. . 3
⊢
((rank‘𝐴)
∈ On ↔ ((rank‘𝐴) = ∅ ∨ ∃𝑥 ∈ On (rank‘𝐴) = suc 𝑥 ∨ ((rank‘𝐴) ∈ V ∧ Lim (rank‘𝐴)))) |
3 | 1, 2 | mpbi 219 |
. 2
⊢
((rank‘𝐴) =
∅ ∨ ∃𝑥
∈ On (rank‘𝐴) =
suc 𝑥 ∨
((rank‘𝐴) ∈ V
∧ Lim (rank‘𝐴))) |
4 | | sdom0 7977 |
. . . 4
⊢ ¬
𝐴 ≺
∅ |
5 | | fveq2 6103 |
. . . . . 6
⊢
((rank‘𝐴) =
∅ → (cf‘(rank‘𝐴)) = (cf‘∅)) |
6 | | cf0 8956 |
. . . . . 6
⊢
(cf‘∅) = ∅ |
7 | 5, 6 | syl6eq 2660 |
. . . . 5
⊢
((rank‘𝐴) =
∅ → (cf‘(rank‘𝐴)) = ∅) |
8 | 7 | breq2d 4595 |
. . . 4
⊢
((rank‘𝐴) =
∅ → (𝐴 ≺
(cf‘(rank‘𝐴))
↔ 𝐴 ≺
∅)) |
9 | 4, 8 | mtbiri 316 |
. . 3
⊢
((rank‘𝐴) =
∅ → ¬ 𝐴
≺ (cf‘(rank‘𝐴))) |
10 | | fveq2 6103 |
. . . . . . 7
⊢
((rank‘𝐴) =
suc 𝑥 →
(cf‘(rank‘𝐴)) =
(cf‘suc 𝑥)) |
11 | | cfsuc 8962 |
. . . . . . 7
⊢ (𝑥 ∈ On → (cf‘suc
𝑥) =
1𝑜) |
12 | 10, 11 | sylan9eqr 2666 |
. . . . . 6
⊢ ((𝑥 ∈ On ∧
(rank‘𝐴) = suc 𝑥) →
(cf‘(rank‘𝐴)) =
1𝑜) |
13 | | nsuceq0 5722 |
. . . . . . . . 9
⊢ suc 𝑥 ≠ ∅ |
14 | | neeq1 2844 |
. . . . . . . . 9
⊢
((rank‘𝐴) =
suc 𝑥 →
((rank‘𝐴) ≠
∅ ↔ suc 𝑥 ≠
∅)) |
15 | 13, 14 | mpbiri 247 |
. . . . . . . 8
⊢
((rank‘𝐴) =
suc 𝑥 →
(rank‘𝐴) ≠
∅) |
16 | | fveq2 6103 |
. . . . . . . . . . 11
⊢ (𝐴 = ∅ →
(rank‘𝐴) =
(rank‘∅)) |
17 | | 0elon 5695 |
. . . . . . . . . . . . 13
⊢ ∅
∈ On |
18 | | r1fnon 8513 |
. . . . . . . . . . . . . 14
⊢
𝑅1 Fn On |
19 | | fndm 5904 |
. . . . . . . . . . . . . 14
⊢
(𝑅1 Fn On → dom 𝑅1 =
On) |
20 | 18, 19 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢ dom
𝑅1 = On |
21 | 17, 20 | eleqtrri 2687 |
. . . . . . . . . . . 12
⊢ ∅
∈ dom 𝑅1 |
22 | | rankonid 8575 |
. . . . . . . . . . . 12
⊢ (∅
∈ dom 𝑅1 ↔ (rank‘∅) =
∅) |
23 | 21, 22 | mpbi 219 |
. . . . . . . . . . 11
⊢
(rank‘∅) = ∅ |
24 | 16, 23 | syl6eq 2660 |
. . . . . . . . . 10
⊢ (𝐴 = ∅ →
(rank‘𝐴) =
∅) |
25 | 24 | necon3i 2814 |
. . . . . . . . 9
⊢
((rank‘𝐴) ≠
∅ → 𝐴 ≠
∅) |
26 | | rankvaln 8545 |
. . . . . . . . . . 11
⊢ (¬
𝐴 ∈ ∪ (𝑅1 “ On) →
(rank‘𝐴) =
∅) |
27 | 26 | necon1ai 2809 |
. . . . . . . . . 10
⊢
((rank‘𝐴) ≠
∅ → 𝐴 ∈
∪ (𝑅1 “
On)) |
28 | | breq2 4587 |
. . . . . . . . . . 11
⊢ (𝑦 = 𝐴 → (1𝑜 ≼ 𝑦 ↔ 1𝑜
≼ 𝐴)) |
29 | | neeq1 2844 |
. . . . . . . . . . 11
⊢ (𝑦 = 𝐴 → (𝑦 ≠ ∅ ↔ 𝐴 ≠ ∅)) |
30 | | 0sdom1dom 8043 |
. . . . . . . . . . . 12
⊢ (∅
≺ 𝑦 ↔
1𝑜 ≼ 𝑦) |
31 | | vex 3176 |
. . . . . . . . . . . . 13
⊢ 𝑦 ∈ V |
32 | 31 | 0sdom 7976 |
. . . . . . . . . . . 12
⊢ (∅
≺ 𝑦 ↔ 𝑦 ≠ ∅) |
33 | 30, 32 | bitr3i 265 |
. . . . . . . . . . 11
⊢
(1𝑜 ≼ 𝑦 ↔ 𝑦 ≠ ∅) |
34 | 28, 29, 33 | vtoclbg 3240 |
. . . . . . . . . 10
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) →
(1𝑜 ≼ 𝐴 ↔ 𝐴 ≠ ∅)) |
35 | 27, 34 | syl 17 |
. . . . . . . . 9
⊢
((rank‘𝐴) ≠
∅ → (1𝑜 ≼ 𝐴 ↔ 𝐴 ≠ ∅)) |
36 | 25, 35 | mpbird 246 |
. . . . . . . 8
⊢
((rank‘𝐴) ≠
∅ → 1𝑜 ≼ 𝐴) |
37 | 15, 36 | syl 17 |
. . . . . . 7
⊢
((rank‘𝐴) =
suc 𝑥 →
1𝑜 ≼ 𝐴) |
38 | 37 | adantl 481 |
. . . . . 6
⊢ ((𝑥 ∈ On ∧
(rank‘𝐴) = suc 𝑥) → 1𝑜
≼ 𝐴) |
39 | 12, 38 | eqbrtrd 4605 |
. . . . 5
⊢ ((𝑥 ∈ On ∧
(rank‘𝐴) = suc 𝑥) →
(cf‘(rank‘𝐴))
≼ 𝐴) |
40 | 39 | rexlimiva 3010 |
. . . 4
⊢
(∃𝑥 ∈ On
(rank‘𝐴) = suc 𝑥 →
(cf‘(rank‘𝐴))
≼ 𝐴) |
41 | | domnsym 7971 |
. . . 4
⊢
((cf‘(rank‘𝐴)) ≼ 𝐴 → ¬ 𝐴 ≺ (cf‘(rank‘𝐴))) |
42 | 40, 41 | syl 17 |
. . 3
⊢
(∃𝑥 ∈ On
(rank‘𝐴) = suc 𝑥 → ¬ 𝐴 ≺ (cf‘(rank‘𝐴))) |
43 | | nlim0 5700 |
. . . . . . . . . . . . . . . . 17
⊢ ¬
Lim ∅ |
44 | | limeq 5652 |
. . . . . . . . . . . . . . . . 17
⊢
((rank‘𝐴) =
∅ → (Lim (rank‘𝐴) ↔ Lim ∅)) |
45 | 43, 44 | mtbiri 316 |
. . . . . . . . . . . . . . . 16
⊢
((rank‘𝐴) =
∅ → ¬ Lim (rank‘𝐴)) |
46 | 26, 45 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (¬
𝐴 ∈ ∪ (𝑅1 “ On) → ¬ Lim
(rank‘𝐴)) |
47 | 46 | con4i 112 |
. . . . . . . . . . . . . 14
⊢ (Lim
(rank‘𝐴) → 𝐴 ∈ ∪ (𝑅1 “ On)) |
48 | | r1elssi 8551 |
. . . . . . . . . . . . . 14
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ ∪ (𝑅1 “ On)) |
49 | 47, 48 | syl 17 |
. . . . . . . . . . . . 13
⊢ (Lim
(rank‘𝐴) → 𝐴 ⊆ ∪ (𝑅1 “ On)) |
50 | 49 | sselda 3568 |
. . . . . . . . . . . 12
⊢ ((Lim
(rank‘𝐴) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ∪
(𝑅1 “ On)) |
51 | | ranksnb 8573 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ∪ (𝑅1 “ On) →
(rank‘{𝑥}) = suc
(rank‘𝑥)) |
52 | 50, 51 | syl 17 |
. . . . . . . . . . 11
⊢ ((Lim
(rank‘𝐴) ∧ 𝑥 ∈ 𝐴) → (rank‘{𝑥}) = suc (rank‘𝑥)) |
53 | | rankelb 8570 |
. . . . . . . . . . . . . 14
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝑥 ∈ 𝐴 → (rank‘𝑥) ∈ (rank‘𝐴))) |
54 | 47, 53 | syl 17 |
. . . . . . . . . . . . 13
⊢ (Lim
(rank‘𝐴) →
(𝑥 ∈ 𝐴 → (rank‘𝑥) ∈ (rank‘𝐴))) |
55 | | limsuc 6941 |
. . . . . . . . . . . . 13
⊢ (Lim
(rank‘𝐴) →
((rank‘𝑥) ∈
(rank‘𝐴) ↔ suc
(rank‘𝑥) ∈
(rank‘𝐴))) |
56 | 54, 55 | sylibd 228 |
. . . . . . . . . . . 12
⊢ (Lim
(rank‘𝐴) →
(𝑥 ∈ 𝐴 → suc (rank‘𝑥) ∈ (rank‘𝐴))) |
57 | 56 | imp 444 |
. . . . . . . . . . 11
⊢ ((Lim
(rank‘𝐴) ∧ 𝑥 ∈ 𝐴) → suc (rank‘𝑥) ∈ (rank‘𝐴)) |
58 | 52, 57 | eqeltrd 2688 |
. . . . . . . . . 10
⊢ ((Lim
(rank‘𝐴) ∧ 𝑥 ∈ 𝐴) → (rank‘{𝑥}) ∈ (rank‘𝐴)) |
59 | | eleq1a 2683 |
. . . . . . . . . 10
⊢
((rank‘{𝑥})
∈ (rank‘𝐴)
→ (𝑤 =
(rank‘{𝑥}) →
𝑤 ∈ (rank‘𝐴))) |
60 | 58, 59 | syl 17 |
. . . . . . . . 9
⊢ ((Lim
(rank‘𝐴) ∧ 𝑥 ∈ 𝐴) → (𝑤 = (rank‘{𝑥}) → 𝑤 ∈ (rank‘𝐴))) |
61 | 60 | rexlimdva 3013 |
. . . . . . . 8
⊢ (Lim
(rank‘𝐴) →
(∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥}) → 𝑤 ∈ (rank‘𝐴))) |
62 | 61 | abssdv 3639 |
. . . . . . 7
⊢ (Lim
(rank‘𝐴) →
{𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})} ⊆ (rank‘𝐴)) |
63 | | snex 4835 |
. . . . . . . . . . . . 13
⊢ {𝑥} ∈ V |
64 | 63 | dfiun2 4490 |
. . . . . . . . . . . 12
⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} |
65 | | iunid 4511 |
. . . . . . . . . . . 12
⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴 |
66 | 64, 65 | eqtr3i 2634 |
. . . . . . . . . . 11
⊢ ∪ {𝑦
∣ ∃𝑥 ∈
𝐴 𝑦 = {𝑥}} = 𝐴 |
67 | 66 | fveq2i 6106 |
. . . . . . . . . 10
⊢
(rank‘∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}}) = (rank‘𝐴) |
68 | 48 | sselda 3568 |
. . . . . . . . . . . . . . 15
⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ∪
(𝑅1 “ On)) |
69 | | snwf 8555 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ ∪ (𝑅1 “ On) → {𝑥} ∈ ∪ (𝑅1 “ On)) |
70 | | eleq1a 2683 |
. . . . . . . . . . . . . . 15
⊢ ({𝑥} ∈ ∪ (𝑅1 “ On) → (𝑦 = {𝑥} → 𝑦 ∈ ∪
(𝑅1 “ On))) |
71 | 68, 69, 70 | 3syl 18 |
. . . . . . . . . . . . . 14
⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝑥 ∈ 𝐴) → (𝑦 = {𝑥} → 𝑦 ∈ ∪
(𝑅1 “ On))) |
72 | 71 | rexlimdva 3013 |
. . . . . . . . . . . . 13
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (∃𝑥 ∈ 𝐴 𝑦 = {𝑥} → 𝑦 ∈ ∪
(𝑅1 “ On))) |
73 | 72 | abssdv 3639 |
. . . . . . . . . . . 12
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} ⊆ ∪
(𝑅1 “ On)) |
74 | | abrexexg 7034 |
. . . . . . . . . . . . 13
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} ∈ V) |
75 | | eleq1 2676 |
. . . . . . . . . . . . . 14
⊢ (𝑧 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} → (𝑧 ∈ ∪
(𝑅1 “ On) ↔ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} ∈ ∪
(𝑅1 “ On))) |
76 | | sseq1 3589 |
. . . . . . . . . . . . . 14
⊢ (𝑧 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} → (𝑧 ⊆ ∪
(𝑅1 “ On) ↔ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} ⊆ ∪
(𝑅1 “ On))) |
77 | | vex 3176 |
. . . . . . . . . . . . . . 15
⊢ 𝑧 ∈ V |
78 | 77 | r1elss 8552 |
. . . . . . . . . . . . . 14
⊢ (𝑧 ∈ ∪ (𝑅1 “ On) ↔ 𝑧 ⊆ ∪ (𝑅1 “ On)) |
79 | 75, 76, 78 | vtoclbg 3240 |
. . . . . . . . . . . . 13
⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} ∈ V → ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} ∈ ∪
(𝑅1 “ On) ↔ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} ⊆ ∪
(𝑅1 “ On))) |
80 | 74, 79 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} ∈ ∪
(𝑅1 “ On) ↔ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} ⊆ ∪
(𝑅1 “ On))) |
81 | 73, 80 | mpbird 246 |
. . . . . . . . . . 11
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} ∈ ∪
(𝑅1 “ On)) |
82 | | rankuni2b 8599 |
. . . . . . . . . . 11
⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} ∈ ∪
(𝑅1 “ On) → (rank‘∪ {𝑦
∣ ∃𝑥 ∈
𝐴 𝑦 = {𝑥}}) = ∪
𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} (rank‘𝑧)) |
83 | 81, 82 | syl 17 |
. . . . . . . . . 10
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) →
(rank‘∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}}) = ∪
𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} (rank‘𝑧)) |
84 | 67, 83 | syl5eqr 2658 |
. . . . . . . . 9
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) →
(rank‘𝐴) = ∪ 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} (rank‘𝑧)) |
85 | | fvex 6113 |
. . . . . . . . . . 11
⊢
(rank‘𝑧)
∈ V |
86 | 85 | dfiun2 4490 |
. . . . . . . . . 10
⊢ ∪ 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} (rank‘𝑧) = ∪ {𝑤 ∣ ∃𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}}𝑤 = (rank‘𝑧)} |
87 | | fveq2 6103 |
. . . . . . . . . . . 12
⊢ (𝑧 = {𝑥} → (rank‘𝑧) = (rank‘{𝑥})) |
88 | 63, 87 | abrexco 6406 |
. . . . . . . . . . 11
⊢ {𝑤 ∣ ∃𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}}𝑤 = (rank‘𝑧)} = {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})} |
89 | 88 | unieqi 4381 |
. . . . . . . . . 10
⊢ ∪ {𝑤
∣ ∃𝑧 ∈
{𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}}𝑤 = (rank‘𝑧)} = ∪ {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})} |
90 | 86, 89 | eqtri 2632 |
. . . . . . . . 9
⊢ ∪ 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} (rank‘𝑧) = ∪ {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})} |
91 | 84, 90 | syl6req 2661 |
. . . . . . . 8
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∪ {𝑤
∣ ∃𝑥 ∈
𝐴 𝑤 = (rank‘{𝑥})} = (rank‘𝐴)) |
92 | 47, 91 | syl 17 |
. . . . . . 7
⊢ (Lim
(rank‘𝐴) → ∪ {𝑤
∣ ∃𝑥 ∈
𝐴 𝑤 = (rank‘{𝑥})} = (rank‘𝐴)) |
93 | | fvex 6113 |
. . . . . . . 8
⊢
(rank‘𝐴)
∈ V |
94 | 93 | cfslb 8971 |
. . . . . . 7
⊢ ((Lim
(rank‘𝐴) ∧ {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})} ⊆ (rank‘𝐴) ∧ ∪ {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})} = (rank‘𝐴)) → (cf‘(rank‘𝐴)) ≼ {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})}) |
95 | 62, 92, 94 | mpd3an23 1418 |
. . . . . 6
⊢ (Lim
(rank‘𝐴) →
(cf‘(rank‘𝐴))
≼ {𝑤 ∣
∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})}) |
96 | | fveq2 6103 |
. . . . . . . . . . 11
⊢ (𝑦 = 𝐴 → (rank‘𝑦) = (rank‘𝐴)) |
97 | 96 | fveq2d 6107 |
. . . . . . . . . 10
⊢ (𝑦 = 𝐴 → (cf‘(rank‘𝑦)) = (cf‘(rank‘𝐴))) |
98 | | breq12 4588 |
. . . . . . . . . 10
⊢ ((𝑦 = 𝐴 ∧ (cf‘(rank‘𝑦)) = (cf‘(rank‘𝐴))) → (𝑦 ≺ (cf‘(rank‘𝑦)) ↔ 𝐴 ≺ (cf‘(rank‘𝐴)))) |
99 | 97, 98 | mpdan 699 |
. . . . . . . . 9
⊢ (𝑦 = 𝐴 → (𝑦 ≺ (cf‘(rank‘𝑦)) ↔ 𝐴 ≺ (cf‘(rank‘𝐴)))) |
100 | | rexeq 3116 |
. . . . . . . . . . 11
⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ 𝑦 𝑤 = (rank‘{𝑥}) ↔ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥}))) |
101 | 100 | abbidv 2728 |
. . . . . . . . . 10
⊢ (𝑦 = 𝐴 → {𝑤 ∣ ∃𝑥 ∈ 𝑦 𝑤 = (rank‘{𝑥})} = {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})}) |
102 | | breq12 4588 |
. . . . . . . . . 10
⊢ (({𝑤 ∣ ∃𝑥 ∈ 𝑦 𝑤 = (rank‘{𝑥})} = {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})} ∧ 𝑦 = 𝐴) → ({𝑤 ∣ ∃𝑥 ∈ 𝑦 𝑤 = (rank‘{𝑥})} ≼ 𝑦 ↔ {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})} ≼ 𝐴)) |
103 | 101, 102 | mpancom 700 |
. . . . . . . . 9
⊢ (𝑦 = 𝐴 → ({𝑤 ∣ ∃𝑥 ∈ 𝑦 𝑤 = (rank‘{𝑥})} ≼ 𝑦 ↔ {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})} ≼ 𝐴)) |
104 | 99, 103 | imbi12d 333 |
. . . . . . . 8
⊢ (𝑦 = 𝐴 → ((𝑦 ≺ (cf‘(rank‘𝑦)) → {𝑤 ∣ ∃𝑥 ∈ 𝑦 𝑤 = (rank‘{𝑥})} ≼ 𝑦) ↔ (𝐴 ≺ (cf‘(rank‘𝐴)) → {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})} ≼ 𝐴))) |
105 | | eqid 2610 |
. . . . . . . . . 10
⊢ (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})) = (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})) |
106 | 105 | rnmpt 5292 |
. . . . . . . . 9
⊢ ran
(𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})) = {𝑤 ∣ ∃𝑥 ∈ 𝑦 𝑤 = (rank‘{𝑥})} |
107 | | cfon 8960 |
. . . . . . . . . . 11
⊢
(cf‘(rank‘𝑦)) ∈ On |
108 | | sdomdom 7869 |
. . . . . . . . . . 11
⊢ (𝑦 ≺
(cf‘(rank‘𝑦))
→ 𝑦 ≼
(cf‘(rank‘𝑦))) |
109 | | ondomen 8743 |
. . . . . . . . . . 11
⊢
(((cf‘(rank‘𝑦)) ∈ On ∧ 𝑦 ≼ (cf‘(rank‘𝑦))) → 𝑦 ∈ dom card) |
110 | 107, 108,
109 | sylancr 694 |
. . . . . . . . . 10
⊢ (𝑦 ≺
(cf‘(rank‘𝑦))
→ 𝑦 ∈ dom
card) |
111 | | fvex 6113 |
. . . . . . . . . . . 12
⊢
(rank‘{𝑥})
∈ V |
112 | 111, 105 | fnmpti 5935 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})) Fn 𝑦 |
113 | | dffn4 6034 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})) Fn 𝑦 ↔ (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})):𝑦–onto→ran (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥}))) |
114 | 112, 113 | mpbi 219 |
. . . . . . . . . 10
⊢ (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})):𝑦–onto→ran (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})) |
115 | | fodomnum 8763 |
. . . . . . . . . 10
⊢ (𝑦 ∈ dom card → ((𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})):𝑦–onto→ran (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})) → ran (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})) ≼ 𝑦)) |
116 | 110, 114,
115 | mpisyl 21 |
. . . . . . . . 9
⊢ (𝑦 ≺
(cf‘(rank‘𝑦))
→ ran (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})) ≼ 𝑦) |
117 | 106, 116 | syl5eqbrr 4619 |
. . . . . . . 8
⊢ (𝑦 ≺
(cf‘(rank‘𝑦))
→ {𝑤 ∣
∃𝑥 ∈ 𝑦 𝑤 = (rank‘{𝑥})} ≼ 𝑦) |
118 | 104, 117 | vtoclg 3239 |
. . . . . . 7
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝐴 ≺
(cf‘(rank‘𝐴))
→ {𝑤 ∣
∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})} ≼ 𝐴)) |
119 | 47, 118 | syl 17 |
. . . . . 6
⊢ (Lim
(rank‘𝐴) →
(𝐴 ≺
(cf‘(rank‘𝐴))
→ {𝑤 ∣
∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})} ≼ 𝐴)) |
120 | | domtr 7895 |
. . . . . . 7
⊢
(((cf‘(rank‘𝐴)) ≼ {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})} ∧ {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})} ≼ 𝐴) → (cf‘(rank‘𝐴)) ≼ 𝐴) |
121 | 120, 41 | syl 17 |
. . . . . 6
⊢
(((cf‘(rank‘𝐴)) ≼ {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})} ∧ {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})} ≼ 𝐴) → ¬ 𝐴 ≺ (cf‘(rank‘𝐴))) |
122 | 95, 119, 121 | syl6an 566 |
. . . . 5
⊢ (Lim
(rank‘𝐴) →
(𝐴 ≺
(cf‘(rank‘𝐴))
→ ¬ 𝐴 ≺
(cf‘(rank‘𝐴)))) |
123 | 122 | pm2.01d 180 |
. . . 4
⊢ (Lim
(rank‘𝐴) → ¬
𝐴 ≺
(cf‘(rank‘𝐴))) |
124 | 123 | adantl 481 |
. . 3
⊢
(((rank‘𝐴)
∈ V ∧ Lim (rank‘𝐴)) → ¬ 𝐴 ≺ (cf‘(rank‘𝐴))) |
125 | 9, 42, 124 | 3jaoi 1383 |
. 2
⊢
(((rank‘𝐴) =
∅ ∨ ∃𝑥
∈ On (rank‘𝐴) =
suc 𝑥 ∨
((rank‘𝐴) ∈ V
∧ Lim (rank‘𝐴)))
→ ¬ 𝐴 ≺
(cf‘(rank‘𝐴))) |
126 | 3, 125 | ax-mp 5 |
1
⊢ ¬
𝐴 ≺
(cf‘(rank‘𝐴)) |