Step | Hyp | Ref
| Expression |
1 | | rankon 8935 |
. . 3
⊢
(rank‘𝐴)
∈ On |
2 | | onzsl 7307 |
. . 3
⊢
((rank‘𝐴)
∈ On ↔ ((rank‘𝐴) = ∅ ∨ ∃𝑥 ∈ On (rank‘𝐴) = suc 𝑥 ∨ ((rank‘𝐴) ∈ V ∧ Lim (rank‘𝐴)))) |
3 | 1, 2 | mpbi 222 |
. 2
⊢
((rank‘𝐴) =
∅ ∨ ∃𝑥
∈ On (rank‘𝐴) =
suc 𝑥 ∨
((rank‘𝐴) ∈ V
∧ Lim (rank‘𝐴))) |
4 | | sdom0 8361 |
. . . 4
⊢ ¬
𝐴 ≺
∅ |
5 | | fveq2 6433 |
. . . . . 6
⊢
((rank‘𝐴) =
∅ → (cf‘(rank‘𝐴)) = (cf‘∅)) |
6 | | cf0 9388 |
. . . . . 6
⊢
(cf‘∅) = ∅ |
7 | 5, 6 | syl6eq 2877 |
. . . . 5
⊢
((rank‘𝐴) =
∅ → (cf‘(rank‘𝐴)) = ∅) |
8 | 7 | breq2d 4885 |
. . . 4
⊢
((rank‘𝐴) =
∅ → (𝐴 ≺
(cf‘(rank‘𝐴))
↔ 𝐴 ≺
∅)) |
9 | 4, 8 | mtbiri 319 |
. . 3
⊢
((rank‘𝐴) =
∅ → ¬ 𝐴
≺ (cf‘(rank‘𝐴))) |
10 | | fveq2 6433 |
. . . . . . 7
⊢
((rank‘𝐴) =
suc 𝑥 →
(cf‘(rank‘𝐴)) =
(cf‘suc 𝑥)) |
11 | | cfsuc 9394 |
. . . . . . 7
⊢ (𝑥 ∈ On → (cf‘suc
𝑥) =
1o) |
12 | 10, 11 | sylan9eqr 2883 |
. . . . . 6
⊢ ((𝑥 ∈ On ∧
(rank‘𝐴) = suc 𝑥) →
(cf‘(rank‘𝐴)) =
1o) |
13 | | nsuceq0 6043 |
. . . . . . . . 9
⊢ suc 𝑥 ≠ ∅ |
14 | | neeq1 3061 |
. . . . . . . . 9
⊢
((rank‘𝐴) =
suc 𝑥 →
((rank‘𝐴) ≠
∅ ↔ suc 𝑥 ≠
∅)) |
15 | 13, 14 | mpbiri 250 |
. . . . . . . 8
⊢
((rank‘𝐴) =
suc 𝑥 →
(rank‘𝐴) ≠
∅) |
16 | | fveq2 6433 |
. . . . . . . . . . 11
⊢ (𝐴 = ∅ →
(rank‘𝐴) =
(rank‘∅)) |
17 | | 0elon 6016 |
. . . . . . . . . . . . 13
⊢ ∅
∈ On |
18 | | r1fnon 8907 |
. . . . . . . . . . . . . 14
⊢
𝑅1 Fn On |
19 | | fndm 6223 |
. . . . . . . . . . . . . 14
⊢
(𝑅1 Fn On → dom 𝑅1 =
On) |
20 | 18, 19 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢ dom
𝑅1 = On |
21 | 17, 20 | eleqtrri 2905 |
. . . . . . . . . . . 12
⊢ ∅
∈ dom 𝑅1 |
22 | | rankonid 8969 |
. . . . . . . . . . . 12
⊢ (∅
∈ dom 𝑅1 ↔ (rank‘∅) =
∅) |
23 | 21, 22 | mpbi 222 |
. . . . . . . . . . 11
⊢
(rank‘∅) = ∅ |
24 | 16, 23 | syl6eq 2877 |
. . . . . . . . . 10
⊢ (𝐴 = ∅ →
(rank‘𝐴) =
∅) |
25 | 24 | necon3i 3031 |
. . . . . . . . 9
⊢
((rank‘𝐴) ≠
∅ → 𝐴 ≠
∅) |
26 | | rankvaln 8939 |
. . . . . . . . . . 11
⊢ (¬
𝐴 ∈ ∪ (𝑅1 “ On) →
(rank‘𝐴) =
∅) |
27 | 26 | necon1ai 3026 |
. . . . . . . . . 10
⊢
((rank‘𝐴) ≠
∅ → 𝐴 ∈
∪ (𝑅1 “
On)) |
28 | | breq2 4877 |
. . . . . . . . . . 11
⊢ (𝑦 = 𝐴 → (1o ≼ 𝑦 ↔ 1o ≼
𝐴)) |
29 | | neeq1 3061 |
. . . . . . . . . . 11
⊢ (𝑦 = 𝐴 → (𝑦 ≠ ∅ ↔ 𝐴 ≠ ∅)) |
30 | | 0sdom1dom 8427 |
. . . . . . . . . . . 12
⊢ (∅
≺ 𝑦 ↔
1o ≼ 𝑦) |
31 | | vex 3417 |
. . . . . . . . . . . . 13
⊢ 𝑦 ∈ V |
32 | 31 | 0sdom 8360 |
. . . . . . . . . . . 12
⊢ (∅
≺ 𝑦 ↔ 𝑦 ≠ ∅) |
33 | 30, 32 | bitr3i 269 |
. . . . . . . . . . 11
⊢
(1o ≼ 𝑦 ↔ 𝑦 ≠ ∅) |
34 | 28, 29, 33 | vtoclbg 3483 |
. . . . . . . . . 10
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (1o
≼ 𝐴 ↔ 𝐴 ≠ ∅)) |
35 | 27, 34 | syl 17 |
. . . . . . . . 9
⊢
((rank‘𝐴) ≠
∅ → (1o ≼ 𝐴 ↔ 𝐴 ≠ ∅)) |
36 | 25, 35 | mpbird 249 |
. . . . . . . 8
⊢
((rank‘𝐴) ≠
∅ → 1o ≼ 𝐴) |
37 | 15, 36 | syl 17 |
. . . . . . 7
⊢
((rank‘𝐴) =
suc 𝑥 → 1o
≼ 𝐴) |
38 | 37 | adantl 475 |
. . . . . 6
⊢ ((𝑥 ∈ On ∧
(rank‘𝐴) = suc 𝑥) → 1o ≼
𝐴) |
39 | 12, 38 | eqbrtrd 4895 |
. . . . 5
⊢ ((𝑥 ∈ On ∧
(rank‘𝐴) = suc 𝑥) →
(cf‘(rank‘𝐴))
≼ 𝐴) |
40 | 39 | rexlimiva 3237 |
. . . 4
⊢
(∃𝑥 ∈ On
(rank‘𝐴) = suc 𝑥 →
(cf‘(rank‘𝐴))
≼ 𝐴) |
41 | | domnsym 8355 |
. . . 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 6021 |
. . . . . . . . . . . . . . . . 17
⊢ ¬
Lim ∅ |
44 | | limeq 5975 |
. . . . . . . . . . . . . . . . 17
⊢
((rank‘𝐴) =
∅ → (Lim (rank‘𝐴) ↔ Lim ∅)) |
45 | 43, 44 | mtbiri 319 |
. . . . . . . . . . . . . . . 16
⊢
((rank‘𝐴) =
∅ → ¬ Lim (rank‘𝐴)) |
46 | 26, 45 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (¬
𝐴 ∈ ∪ (𝑅1 “ On) → ¬ Lim
(rank‘𝐴)) |
47 | 46 | con4i 114 |
. . . . . . . . . . . . . 14
⊢ (Lim
(rank‘𝐴) → 𝐴 ∈ ∪ (𝑅1 “ On)) |
48 | | r1elssi 8945 |
. . . . . . . . . . . . . 14
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ ∪ (𝑅1 “ On)) |
49 | 47, 48 | syl 17 |
. . . . . . . . . . . . 13
⊢ (Lim
(rank‘𝐴) → 𝐴 ⊆ ∪ (𝑅1 “ On)) |
50 | 49 | sselda 3827 |
. . . . . . . . . . . 12
⊢ ((Lim
(rank‘𝐴) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ∪
(𝑅1 “ On)) |
51 | | ranksnb 8967 |
. . . . . . . . . . . 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 8964 |
. . . . . . . . . . . . . 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 7310 |
. . . . . . . . . . . . 13
⊢ (Lim
(rank‘𝐴) →
((rank‘𝑥) ∈
(rank‘𝐴) ↔ suc
(rank‘𝑥) ∈
(rank‘𝐴))) |
56 | 54, 55 | sylibd 231 |
. . . . . . . . . . . 12
⊢ (Lim
(rank‘𝐴) →
(𝑥 ∈ 𝐴 → suc (rank‘𝑥) ∈ (rank‘𝐴))) |
57 | 56 | imp 397 |
. . . . . . . . . . 11
⊢ ((Lim
(rank‘𝐴) ∧ 𝑥 ∈ 𝐴) → suc (rank‘𝑥) ∈ (rank‘𝐴)) |
58 | 52, 57 | eqeltrd 2906 |
. . . . . . . . . 10
⊢ ((Lim
(rank‘𝐴) ∧ 𝑥 ∈ 𝐴) → (rank‘{𝑥}) ∈ (rank‘𝐴)) |
59 | | eleq1a 2901 |
. . . . . . . . . 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 3240 |
. . . . . . . 8
⊢ (Lim
(rank‘𝐴) →
(∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥}) → 𝑤 ∈ (rank‘𝐴))) |
62 | 61 | abssdv 3901 |
. . . . . . 7
⊢ (Lim
(rank‘𝐴) →
{𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})} ⊆ (rank‘𝐴)) |
63 | | snex 5129 |
. . . . . . . . . . . . 13
⊢ {𝑥} ∈ V |
64 | 63 | dfiun2 4774 |
. . . . . . . . . . . 12
⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} |
65 | | iunid 4795 |
. . . . . . . . . . . 12
⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴 |
66 | 64, 65 | eqtr3i 2851 |
. . . . . . . . . . 11
⊢ ∪ {𝑦
∣ ∃𝑥 ∈
𝐴 𝑦 = {𝑥}} = 𝐴 |
67 | 66 | fveq2i 6436 |
. . . . . . . . . 10
⊢
(rank‘∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}}) = (rank‘𝐴) |
68 | 48 | sselda 3827 |
. . . . . . . . . . . . . . 15
⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ∪
(𝑅1 “ On)) |
69 | | snwf 8949 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ ∪ (𝑅1 “ On) → {𝑥} ∈ ∪ (𝑅1 “ On)) |
70 | | eleq1a 2901 |
. . . . . . . . . . . . . . 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 3240 |
. . . . . . . . . . . . 13
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (∃𝑥 ∈ 𝐴 𝑦 = {𝑥} → 𝑦 ∈ ∪
(𝑅1 “ On))) |
73 | 72 | abssdv 3901 |
. . . . . . . . . . . 12
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} ⊆ ∪
(𝑅1 “ On)) |
74 | | abrexexg 7402 |
. . . . . . . . . . . . 13
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} ∈ V) |
75 | | eleq1 2894 |
. . . . . . . . . . . . . 14
⊢ (𝑧 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} → (𝑧 ∈ ∪
(𝑅1 “ On) ↔ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} ∈ ∪
(𝑅1 “ On))) |
76 | | sseq1 3851 |
. . . . . . . . . . . . . 14
⊢ (𝑧 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} → (𝑧 ⊆ ∪
(𝑅1 “ On) ↔ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} ⊆ ∪
(𝑅1 “ On))) |
77 | | vex 3417 |
. . . . . . . . . . . . . . 15
⊢ 𝑧 ∈ V |
78 | 77 | r1elss 8946 |
. . . . . . . . . . . . . 14
⊢ (𝑧 ∈ ∪ (𝑅1 “ On) ↔ 𝑧 ⊆ ∪ (𝑅1 “ On)) |
79 | 75, 76, 78 | vtoclbg 3483 |
. . . . . . . . . . . . 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 249 |
. . . . . . . . . . 11
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} ∈ ∪
(𝑅1 “ On)) |
82 | | rankuni2b 8993 |
. . . . . . . . . . 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 2875 |
. . . . . . . . 9
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) →
(rank‘𝐴) = ∪ 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} (rank‘𝑧)) |
85 | | fvex 6446 |
. . . . . . . . . . 11
⊢
(rank‘𝑧)
∈ V |
86 | 85 | dfiun2 4774 |
. . . . . . . . . 10
⊢ ∪ 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} (rank‘𝑧) = ∪ {𝑤 ∣ ∃𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}}𝑤 = (rank‘𝑧)} |
87 | | fveq2 6433 |
. . . . . . . . . . . 12
⊢ (𝑧 = {𝑥} → (rank‘𝑧) = (rank‘{𝑥})) |
88 | 63, 87 | abrexco 6757 |
. . . . . . . . . . 11
⊢ {𝑤 ∣ ∃𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}}𝑤 = (rank‘𝑧)} = {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})} |
89 | 88 | unieqi 4667 |
. . . . . . . . . 10
⊢ ∪ {𝑤
∣ ∃𝑧 ∈
{𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}}𝑤 = (rank‘𝑧)} = ∪ {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})} |
90 | 86, 89 | eqtri 2849 |
. . . . . . . . 9
⊢ ∪ 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} (rank‘𝑧) = ∪ {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})} |
91 | 84, 90 | syl6req 2878 |
. . . . . . . 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 6446 |
. . . . . . . 8
⊢
(rank‘𝐴)
∈ V |
94 | 93 | cfslb 9403 |
. . . . . . 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 1593 |
. . . . . 6
⊢ (Lim
(rank‘𝐴) →
(cf‘(rank‘𝐴))
≼ {𝑤 ∣
∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})}) |
96 | | 2fveq3 6438 |
. . . . . . . . . 10
⊢ (𝑦 = 𝐴 → (cf‘(rank‘𝑦)) = (cf‘(rank‘𝐴))) |
97 | | breq12 4878 |
. . . . . . . . . 10
⊢ ((𝑦 = 𝐴 ∧ (cf‘(rank‘𝑦)) = (cf‘(rank‘𝐴))) → (𝑦 ≺ (cf‘(rank‘𝑦)) ↔ 𝐴 ≺ (cf‘(rank‘𝐴)))) |
98 | 96, 97 | mpdan 680 |
. . . . . . . . 9
⊢ (𝑦 = 𝐴 → (𝑦 ≺ (cf‘(rank‘𝑦)) ↔ 𝐴 ≺ (cf‘(rank‘𝐴)))) |
99 | | rexeq 3351 |
. . . . . . . . . . 11
⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ 𝑦 𝑤 = (rank‘{𝑥}) ↔ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥}))) |
100 | 99 | abbidv 2946 |
. . . . . . . . . 10
⊢ (𝑦 = 𝐴 → {𝑤 ∣ ∃𝑥 ∈ 𝑦 𝑤 = (rank‘{𝑥})} = {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})}) |
101 | | breq12 4878 |
. . . . . . . . . 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)) |
102 | 100, 101 | mpancom 681 |
. . . . . . . . 9
⊢ (𝑦 = 𝐴 → ({𝑤 ∣ ∃𝑥 ∈ 𝑦 𝑤 = (rank‘{𝑥})} ≼ 𝑦 ↔ {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})} ≼ 𝐴)) |
103 | 98, 102 | imbi12d 336 |
. . . . . . . 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))) |
104 | | eqid 2825 |
. . . . . . . . . 10
⊢ (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})) = (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})) |
105 | 104 | rnmpt 5604 |
. . . . . . . . 9
⊢ ran
(𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})) = {𝑤 ∣ ∃𝑥 ∈ 𝑦 𝑤 = (rank‘{𝑥})} |
106 | | cfon 9392 |
. . . . . . . . . . 11
⊢
(cf‘(rank‘𝑦)) ∈ On |
107 | | sdomdom 8250 |
. . . . . . . . . . 11
⊢ (𝑦 ≺
(cf‘(rank‘𝑦))
→ 𝑦 ≼
(cf‘(rank‘𝑦))) |
108 | | ondomen 9173 |
. . . . . . . . . . 11
⊢
(((cf‘(rank‘𝑦)) ∈ On ∧ 𝑦 ≼ (cf‘(rank‘𝑦))) → 𝑦 ∈ dom card) |
109 | 106, 107,
108 | sylancr 583 |
. . . . . . . . . 10
⊢ (𝑦 ≺
(cf‘(rank‘𝑦))
→ 𝑦 ∈ dom
card) |
110 | | fvex 6446 |
. . . . . . . . . . . 12
⊢
(rank‘{𝑥})
∈ V |
111 | 110, 104 | fnmpti 6255 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})) Fn 𝑦 |
112 | | dffn4 6359 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})) Fn 𝑦 ↔ (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})):𝑦–onto→ran (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥}))) |
113 | 111, 112 | mpbi 222 |
. . . . . . . . . 10
⊢ (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})):𝑦–onto→ran (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})) |
114 | | fodomnum 9193 |
. . . . . . . . . 10
⊢ (𝑦 ∈ dom card → ((𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})):𝑦–onto→ran (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})) → ran (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})) ≼ 𝑦)) |
115 | 109, 113,
114 | mpisyl 21 |
. . . . . . . . 9
⊢ (𝑦 ≺
(cf‘(rank‘𝑦))
→ ran (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})) ≼ 𝑦) |
116 | 105, 115 | syl5eqbrr 4909 |
. . . . . . . 8
⊢ (𝑦 ≺
(cf‘(rank‘𝑦))
→ {𝑤 ∣
∃𝑥 ∈ 𝑦 𝑤 = (rank‘{𝑥})} ≼ 𝑦) |
117 | 103, 116 | vtoclg 3482 |
. . . . . . 7
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝐴 ≺
(cf‘(rank‘𝐴))
→ {𝑤 ∣
∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})} ≼ 𝐴)) |
118 | 47, 117 | syl 17 |
. . . . . 6
⊢ (Lim
(rank‘𝐴) →
(𝐴 ≺
(cf‘(rank‘𝐴))
→ {𝑤 ∣
∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})} ≼ 𝐴)) |
119 | | domtr 8275 |
. . . . . . 7
⊢
(((cf‘(rank‘𝐴)) ≼ {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})} ∧ {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})} ≼ 𝐴) → (cf‘(rank‘𝐴)) ≼ 𝐴) |
120 | 119, 41 | syl 17 |
. . . . . 6
⊢
(((cf‘(rank‘𝐴)) ≼ {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})} ∧ {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})} ≼ 𝐴) → ¬ 𝐴 ≺ (cf‘(rank‘𝐴))) |
121 | 95, 118, 120 | syl6an 676 |
. . . . 5
⊢ (Lim
(rank‘𝐴) →
(𝐴 ≺
(cf‘(rank‘𝐴))
→ ¬ 𝐴 ≺
(cf‘(rank‘𝐴)))) |
122 | 121 | pm2.01d 182 |
. . . 4
⊢ (Lim
(rank‘𝐴) → ¬
𝐴 ≺
(cf‘(rank‘𝐴))) |
123 | 122 | adantl 475 |
. . 3
⊢
(((rank‘𝐴)
∈ V ∧ Lim (rank‘𝐴)) → ¬ 𝐴 ≺ (cf‘(rank‘𝐴))) |
124 | 9, 42, 123 | 3jaoi 1558 |
. 2
⊢
(((rank‘𝐴) =
∅ ∨ ∃𝑥
∈ On (rank‘𝐴) =
suc 𝑥 ∨
((rank‘𝐴) ∈ V
∧ Lim (rank‘𝐴)))
→ ¬ 𝐴 ≺
(cf‘(rank‘𝐴))) |
125 | 3, 124 | ax-mp 5 |
1
⊢ ¬
𝐴 ≺
(cf‘(rank‘𝐴)) |