Proof of Theorem cnfcom3lem
Step | Hyp | Ref
| Expression |
1 | | cnfcom.w |
. . 3
⊢ 𝑊 = (𝐺‘∪ dom
𝐺) |
2 | | cnfcom.a |
. . . 4
⊢ (𝜑 → 𝐴 ∈ On) |
3 | | suppssdm 7195 |
. . . . . 6
⊢ (𝐹 supp ∅) ⊆ dom 𝐹 |
4 | | cnfcom.f |
. . . . . . . . . 10
⊢ 𝐹 = (◡(ω CNF 𝐴)‘𝐵) |
5 | | cnfcom.s |
. . . . . . . . . . . . 13
⊢ 𝑆 = dom (ω CNF 𝐴) |
6 | | omelon 8426 |
. . . . . . . . . . . . . 14
⊢ ω
∈ On |
7 | 6 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ω ∈
On) |
8 | 5, 7, 2 | cantnff1o 8476 |
. . . . . . . . . . . 12
⊢ (𝜑 → (ω CNF 𝐴):𝑆–1-1-onto→(ω ↑𝑜 𝐴)) |
9 | | f1ocnv 6062 |
. . . . . . . . . . . 12
⊢ ((ω
CNF 𝐴):𝑆–1-1-onto→(ω ↑𝑜 𝐴) → ◡(ω CNF 𝐴):(ω ↑𝑜 𝐴)–1-1-onto→𝑆) |
10 | | f1of 6050 |
. . . . . . . . . . . 12
⊢ (◡(ω CNF 𝐴):(ω ↑𝑜 𝐴)–1-1-onto→𝑆 → ◡(ω CNF 𝐴):(ω ↑𝑜 𝐴)⟶𝑆) |
11 | 8, 9, 10 | 3syl 18 |
. . . . . . . . . . 11
⊢ (𝜑 → ◡(ω CNF 𝐴):(ω ↑𝑜 𝐴)⟶𝑆) |
12 | | cnfcom.b |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐵 ∈ (ω ↑𝑜
𝐴)) |
13 | 11, 12 | ffvelrnd 6268 |
. . . . . . . . . 10
⊢ (𝜑 → (◡(ω CNF 𝐴)‘𝐵) ∈ 𝑆) |
14 | 4, 13 | syl5eqel 2692 |
. . . . . . . . 9
⊢ (𝜑 → 𝐹 ∈ 𝑆) |
15 | 5, 7, 2 | cantnfs 8446 |
. . . . . . . . 9
⊢ (𝜑 → (𝐹 ∈ 𝑆 ↔ (𝐹:𝐴⟶ω ∧ 𝐹 finSupp ∅))) |
16 | 14, 15 | mpbid 221 |
. . . . . . . 8
⊢ (𝜑 → (𝐹:𝐴⟶ω ∧ 𝐹 finSupp ∅)) |
17 | 16 | simpld 474 |
. . . . . . 7
⊢ (𝜑 → 𝐹:𝐴⟶ω) |
18 | | fdm 5964 |
. . . . . . 7
⊢ (𝐹:𝐴⟶ω → dom 𝐹 = 𝐴) |
19 | 17, 18 | syl 17 |
. . . . . 6
⊢ (𝜑 → dom 𝐹 = 𝐴) |
20 | 3, 19 | syl5sseq 3616 |
. . . . 5
⊢ (𝜑 → (𝐹 supp ∅) ⊆ 𝐴) |
21 | | ovex 6577 |
. . . . . . . . . . 11
⊢ (𝐹 supp ∅) ∈
V |
22 | | cnfcom.g |
. . . . . . . . . . . 12
⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅)) |
23 | 22 | oion 8324 |
. . . . . . . . . . 11
⊢ ((𝐹 supp ∅) ∈ V →
dom 𝐺 ∈
On) |
24 | 21, 23 | ax-mp 5 |
. . . . . . . . . 10
⊢ dom 𝐺 ∈ On |
25 | 24 | elexi 3186 |
. . . . . . . . 9
⊢ dom 𝐺 ∈ V |
26 | 25 | uniex 6851 |
. . . . . . . 8
⊢ ∪ dom 𝐺 ∈ V |
27 | 26 | sucid 5721 |
. . . . . . 7
⊢ ∪ dom 𝐺 ∈ suc ∪ dom
𝐺 |
28 | | cnfcom.h |
. . . . . . . 8
⊢ 𝐻 =
seq𝜔((𝑘
∈ V, 𝑧 ∈ V
↦ (𝑀
+𝑜 𝑧)),
∅) |
29 | | cnfcom.t |
. . . . . . . 8
⊢ 𝑇 =
seq𝜔((𝑘
∈ V, 𝑓 ∈ V
↦ 𝐾),
∅) |
30 | | cnfcom.m |
. . . . . . . 8
⊢ 𝑀 = ((ω
↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) |
31 | | cnfcom.k |
. . . . . . . 8
⊢ 𝐾 = ((𝑥 ∈ 𝑀 ↦ (dom 𝑓 +𝑜 𝑥)) ∪ ◡(𝑥 ∈ dom 𝑓 ↦ (𝑀 +𝑜 𝑥))) |
32 | | cnfcom3.1 |
. . . . . . . . 9
⊢ (𝜑 → ω ⊆ 𝐵) |
33 | | peano1 6977 |
. . . . . . . . . 10
⊢ ∅
∈ ω |
34 | 33 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → ∅ ∈
ω) |
35 | 32, 34 | sseldd 3569 |
. . . . . . . 8
⊢ (𝜑 → ∅ ∈ 𝐵) |
36 | 5, 2, 12, 4, 22, 28, 29, 30, 31, 1, 35 | cnfcom2lem 8481 |
. . . . . . 7
⊢ (𝜑 → dom 𝐺 = suc ∪ dom
𝐺) |
37 | 27, 36 | syl5eleqr 2695 |
. . . . . 6
⊢ (𝜑 → ∪ dom 𝐺 ∈ dom 𝐺) |
38 | 22 | oif 8318 |
. . . . . . 7
⊢ 𝐺:dom 𝐺⟶(𝐹 supp ∅) |
39 | 38 | ffvelrni 6266 |
. . . . . 6
⊢ (∪ dom 𝐺 ∈ dom 𝐺 → (𝐺‘∪ dom
𝐺) ∈ (𝐹 supp ∅)) |
40 | 37, 39 | syl 17 |
. . . . 5
⊢ (𝜑 → (𝐺‘∪ dom
𝐺) ∈ (𝐹 supp ∅)) |
41 | 20, 40 | sseldd 3569 |
. . . 4
⊢ (𝜑 → (𝐺‘∪ dom
𝐺) ∈ 𝐴) |
42 | | onelon 5665 |
. . . 4
⊢ ((𝐴 ∈ On ∧ (𝐺‘∪ dom 𝐺) ∈ 𝐴) → (𝐺‘∪ dom
𝐺) ∈
On) |
43 | 2, 41, 42 | syl2anc 691 |
. . 3
⊢ (𝜑 → (𝐺‘∪ dom
𝐺) ∈
On) |
44 | 1, 43 | syl5eqel 2692 |
. 2
⊢ (𝜑 → 𝑊 ∈ On) |
45 | | oecl 7504 |
. . . . . . 7
⊢ ((ω
∈ On ∧ 𝐴 ∈
On) → (ω ↑𝑜 𝐴) ∈ On) |
46 | 6, 2, 45 | sylancr 694 |
. . . . . 6
⊢ (𝜑 → (ω
↑𝑜 𝐴) ∈ On) |
47 | | onelon 5665 |
. . . . . 6
⊢
(((ω ↑𝑜 𝐴) ∈ On ∧ 𝐵 ∈ (ω ↑𝑜
𝐴)) → 𝐵 ∈ On) |
48 | 46, 12, 47 | syl2anc 691 |
. . . . 5
⊢ (𝜑 → 𝐵 ∈ On) |
49 | | ontri1 5674 |
. . . . 5
⊢ ((ω
∈ On ∧ 𝐵 ∈
On) → (ω ⊆ 𝐵 ↔ ¬ 𝐵 ∈ ω)) |
50 | 6, 48, 49 | sylancr 694 |
. . . 4
⊢ (𝜑 → (ω ⊆ 𝐵 ↔ ¬ 𝐵 ∈ ω)) |
51 | 32, 50 | mpbid 221 |
. . 3
⊢ (𝜑 → ¬ 𝐵 ∈ ω) |
52 | 4 | fveq2i 6106 |
. . . . . . . 8
⊢ ((ω
CNF 𝐴)‘𝐹) = ((ω CNF 𝐴)‘(◡(ω CNF 𝐴)‘𝐵)) |
53 | | f1ocnvfv2 6433 |
. . . . . . . . 9
⊢
(((ω CNF 𝐴):𝑆–1-1-onto→(ω ↑𝑜 𝐴) ∧ 𝐵 ∈ (ω ↑𝑜
𝐴)) → ((ω CNF
𝐴)‘(◡(ω CNF 𝐴)‘𝐵)) = 𝐵) |
54 | 8, 12, 53 | syl2anc 691 |
. . . . . . . 8
⊢ (𝜑 → ((ω CNF 𝐴)‘(◡(ω CNF 𝐴)‘𝐵)) = 𝐵) |
55 | 52, 54 | syl5eq 2656 |
. . . . . . 7
⊢ (𝜑 → ((ω CNF 𝐴)‘𝐹) = 𝐵) |
56 | 55 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑊 = ∅) → ((ω CNF 𝐴)‘𝐹) = 𝐵) |
57 | 6 | a1i 11 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑊 = ∅) → ω ∈
On) |
58 | 2 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑊 = ∅) → 𝐴 ∈ On) |
59 | 14 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑊 = ∅) → 𝐹 ∈ 𝑆) |
60 | 33 | a1i 11 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑊 = ∅) → ∅ ∈
ω) |
61 | | 1on 7454 |
. . . . . . . . 9
⊢
1𝑜 ∈ On |
62 | 61 | a1i 11 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑊 = ∅) → 1𝑜
∈ On) |
63 | 2, 20 | ssexd 4733 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (𝐹 supp ∅) ∈ V) |
64 | 5, 7, 2, 22, 14 | cantnfcl 8447 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → ( E We (𝐹 supp ∅) ∧ dom 𝐺 ∈ ω)) |
65 | 64 | simpld 474 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → E We (𝐹 supp ∅)) |
66 | 22 | oiiso 8325 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝐹 supp ∅) ∈ V ∧ E
We (𝐹 supp ∅)) →
𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅))) |
67 | 63, 65, 66 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅))) |
68 | 67 | ad2antrr 758 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ (𝐹 supp ∅)) → 𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅))) |
69 | | isof1o 6473 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)) → 𝐺:dom 𝐺–1-1-onto→(𝐹 supp ∅)) |
70 | 68, 69 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ (𝐹 supp ∅)) → 𝐺:dom 𝐺–1-1-onto→(𝐹 supp ∅)) |
71 | | f1ocnv 6062 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐺:dom 𝐺–1-1-onto→(𝐹 supp ∅) → ◡𝐺:(𝐹 supp ∅)–1-1-onto→dom
𝐺) |
72 | | f1of 6050 |
. . . . . . . . . . . . . . . . 17
⊢ (◡𝐺:(𝐹 supp ∅)–1-1-onto→dom
𝐺 → ◡𝐺:(𝐹 supp ∅)⟶dom 𝐺) |
73 | 70, 71, 72 | 3syl 18 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ (𝐹 supp ∅)) → ◡𝐺:(𝐹 supp ∅)⟶dom 𝐺) |
74 | | ffvelrn 6265 |
. . . . . . . . . . . . . . . 16
⊢ ((◡𝐺:(𝐹 supp ∅)⟶dom 𝐺 ∧ 𝑥 ∈ (𝐹 supp ∅)) → (◡𝐺‘𝑥) ∈ dom 𝐺) |
75 | 73, 74 | sylancom 698 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ (𝐹 supp ∅)) → (◡𝐺‘𝑥) ∈ dom 𝐺) |
76 | | elssuni 4403 |
. . . . . . . . . . . . . . 15
⊢ ((◡𝐺‘𝑥) ∈ dom 𝐺 → (◡𝐺‘𝑥) ⊆ ∪ dom
𝐺) |
77 | 75, 76 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ (𝐹 supp ∅)) → (◡𝐺‘𝑥) ⊆ ∪ dom
𝐺) |
78 | | onelon 5665 |
. . . . . . . . . . . . . . . 16
⊢ ((dom
𝐺 ∈ On ∧ (◡𝐺‘𝑥) ∈ dom 𝐺) → (◡𝐺‘𝑥) ∈ On) |
79 | 24, 75, 78 | sylancr 694 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ (𝐹 supp ∅)) → (◡𝐺‘𝑥) ∈ On) |
80 | | onuni 6885 |
. . . . . . . . . . . . . . . 16
⊢ (dom
𝐺 ∈ On → ∪ dom 𝐺 ∈ On) |
81 | 24, 80 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢ ∪ dom 𝐺 ∈ On |
82 | | ontri1 5674 |
. . . . . . . . . . . . . . 15
⊢ (((◡𝐺‘𝑥) ∈ On ∧ ∪ dom 𝐺 ∈ On) → ((◡𝐺‘𝑥) ⊆ ∪ dom
𝐺 ↔ ¬ ∪ dom 𝐺 ∈ (◡𝐺‘𝑥))) |
83 | 79, 81, 82 | sylancl 693 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ (𝐹 supp ∅)) → ((◡𝐺‘𝑥) ⊆ ∪ dom
𝐺 ↔ ¬ ∪ dom 𝐺 ∈ (◡𝐺‘𝑥))) |
84 | 77, 83 | mpbid 221 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ (𝐹 supp ∅)) → ¬ ∪ dom 𝐺 ∈ (◡𝐺‘𝑥)) |
85 | 37 | ad2antrr 758 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ (𝐹 supp ∅)) → ∪ dom 𝐺 ∈ dom 𝐺) |
86 | | isorel 6476 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)) ∧ (∪ dom 𝐺 ∈ dom 𝐺 ∧ (◡𝐺‘𝑥) ∈ dom 𝐺)) → (∪ dom
𝐺 E (◡𝐺‘𝑥) ↔ (𝐺‘∪ dom
𝐺) E (𝐺‘(◡𝐺‘𝑥)))) |
87 | 68, 85, 75, 86 | syl12anc 1316 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ (𝐹 supp ∅)) → (∪ dom 𝐺 E (◡𝐺‘𝑥) ↔ (𝐺‘∪ dom
𝐺) E (𝐺‘(◡𝐺‘𝑥)))) |
88 | | fvex 6113 |
. . . . . . . . . . . . . . . 16
⊢ (◡𝐺‘𝑥) ∈ V |
89 | 88 | epelc 4951 |
. . . . . . . . . . . . . . 15
⊢ (∪ dom 𝐺 E (◡𝐺‘𝑥) ↔ ∪ dom
𝐺 ∈ (◡𝐺‘𝑥)) |
90 | 1 | breq1i 4590 |
. . . . . . . . . . . . . . . 16
⊢ (𝑊 E (𝐺‘(◡𝐺‘𝑥)) ↔ (𝐺‘∪ dom
𝐺) E (𝐺‘(◡𝐺‘𝑥))) |
91 | | fvex 6113 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐺‘(◡𝐺‘𝑥)) ∈ V |
92 | 91 | epelc 4951 |
. . . . . . . . . . . . . . . 16
⊢ (𝑊 E (𝐺‘(◡𝐺‘𝑥)) ↔ 𝑊 ∈ (𝐺‘(◡𝐺‘𝑥))) |
93 | 90, 92 | bitr3i 265 |
. . . . . . . . . . . . . . 15
⊢ ((𝐺‘∪ dom 𝐺) E (𝐺‘(◡𝐺‘𝑥)) ↔ 𝑊 ∈ (𝐺‘(◡𝐺‘𝑥))) |
94 | 87, 89, 93 | 3bitr3g 301 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ (𝐹 supp ∅)) → (∪ dom 𝐺 ∈ (◡𝐺‘𝑥) ↔ 𝑊 ∈ (𝐺‘(◡𝐺‘𝑥)))) |
95 | | simplr 788 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ (𝐹 supp ∅)) → 𝑊 = ∅) |
96 | | f1ocnvfv2 6433 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐺:dom 𝐺–1-1-onto→(𝐹 supp ∅) ∧ 𝑥 ∈ (𝐹 supp ∅)) → (𝐺‘(◡𝐺‘𝑥)) = 𝑥) |
97 | 70, 96 | sylancom 698 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ (𝐹 supp ∅)) → (𝐺‘(◡𝐺‘𝑥)) = 𝑥) |
98 | 95, 97 | eleq12d 2682 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ (𝐹 supp ∅)) → (𝑊 ∈ (𝐺‘(◡𝐺‘𝑥)) ↔ ∅ ∈ 𝑥)) |
99 | 94, 98 | bitrd 267 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ (𝐹 supp ∅)) → (∪ dom 𝐺 ∈ (◡𝐺‘𝑥) ↔ ∅ ∈ 𝑥)) |
100 | 84, 99 | mtbid 313 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ (𝐹 supp ∅)) → ¬ ∅ ∈
𝑥) |
101 | | onss 6882 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐴 ∈ On → 𝐴 ⊆ On) |
102 | 2, 101 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝐴 ⊆ On) |
103 | 20, 102 | sstrd 3578 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝐹 supp ∅) ⊆ On) |
104 | 103 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑊 = ∅) → (𝐹 supp ∅) ⊆ On) |
105 | 104 | sselda 3568 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ (𝐹 supp ∅)) → 𝑥 ∈ On) |
106 | | on0eqel 5762 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ On → (𝑥 = ∅ ∨ ∅ ∈
𝑥)) |
107 | 105, 106 | syl 17 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ (𝐹 supp ∅)) → (𝑥 = ∅ ∨ ∅ ∈ 𝑥)) |
108 | 107 | ord 391 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ (𝐹 supp ∅)) → (¬ 𝑥 = ∅ → ∅ ∈
𝑥)) |
109 | 100, 108 | mt3d 139 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ (𝐹 supp ∅)) → 𝑥 = ∅) |
110 | | el1o 7466 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ 1𝑜
↔ 𝑥 =
∅) |
111 | 109, 110 | sylibr 223 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ 𝑥 ∈ (𝐹 supp ∅)) → 𝑥 ∈
1𝑜) |
112 | 111 | ex 449 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑊 = ∅) → (𝑥 ∈ (𝐹 supp ∅) → 𝑥 ∈
1𝑜)) |
113 | 112 | ssrdv 3574 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑊 = ∅) → (𝐹 supp ∅) ⊆
1𝑜) |
114 | 5, 57, 58, 59, 60, 62, 113 | cantnflt2 8453 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑊 = ∅) → ((ω CNF 𝐴)‘𝐹) ∈ (ω ↑𝑜
1𝑜)) |
115 | | oe1 7511 |
. . . . . . . 8
⊢ (ω
∈ On → (ω ↑𝑜 1𝑜) =
ω) |
116 | 6, 115 | ax-mp 5 |
. . . . . . 7
⊢ (ω
↑𝑜 1𝑜) = ω |
117 | 114, 116 | syl6eleq 2698 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑊 = ∅) → ((ω CNF 𝐴)‘𝐹) ∈ ω) |
118 | 56, 117 | eqeltrrd 2689 |
. . . . 5
⊢ ((𝜑 ∧ 𝑊 = ∅) → 𝐵 ∈ ω) |
119 | 118 | ex 449 |
. . . 4
⊢ (𝜑 → (𝑊 = ∅ → 𝐵 ∈ ω)) |
120 | 119 | necon3bd 2796 |
. . 3
⊢ (𝜑 → (¬ 𝐵 ∈ ω → 𝑊 ≠ ∅)) |
121 | 51, 120 | mpd 15 |
. 2
⊢ (𝜑 → 𝑊 ≠ ∅) |
122 | | dif1o 7467 |
. 2
⊢ (𝑊 ∈ (On ∖
1𝑜) ↔ (𝑊 ∈ On ∧ 𝑊 ≠ ∅)) |
123 | 44, 121, 122 | sylanbrc 695 |
1
⊢ (𝜑 → 𝑊 ∈ (On ∖
1𝑜)) |