Step | Hyp | Ref
| Expression |
1 | | mreexexlem2d.1 |
. . . 4
⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) |
2 | 1 | adantr 485 |
. . 3
⊢ ((𝜑 ∧ 𝐹 = ∅) → 𝐴 ∈ (Moore‘𝑋)) |
3 | | mreexexlem2d.2 |
. . 3
⊢ 𝑁 = (mrCls‘𝐴) |
4 | | mreexexlem2d.3 |
. . 3
⊢ 𝐼 = (mrInd‘𝐴) |
5 | | mreexexlem2d.4 |
. . . 4
⊢ (𝜑 → ∀𝑠 ∈ 𝒫 𝑋∀𝑦 ∈ 𝑋 ∀𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁‘𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧}))) |
6 | 5 | adantr 485 |
. . 3
⊢ ((𝜑 ∧ 𝐹 = ∅) → ∀𝑠 ∈ 𝒫 𝑋∀𝑦 ∈ 𝑋 ∀𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁‘𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧}))) |
7 | | mreexexlem2d.5 |
. . . 4
⊢ (𝜑 → 𝐹 ⊆ (𝑋 ∖ 𝐻)) |
8 | 7 | adantr 485 |
. . 3
⊢ ((𝜑 ∧ 𝐹 = ∅) → 𝐹 ⊆ (𝑋 ∖ 𝐻)) |
9 | | mreexexlem2d.6 |
. . . 4
⊢ (𝜑 → 𝐺 ⊆ (𝑋 ∖ 𝐻)) |
10 | 9 | adantr 485 |
. . 3
⊢ ((𝜑 ∧ 𝐹 = ∅) → 𝐺 ⊆ (𝑋 ∖ 𝐻)) |
11 | | mreexexlem2d.7 |
. . . 4
⊢ (𝜑 → 𝐹 ⊆ (𝑁‘(𝐺 ∪ 𝐻))) |
12 | 11 | adantr 485 |
. . 3
⊢ ((𝜑 ∧ 𝐹 = ∅) → 𝐹 ⊆ (𝑁‘(𝐺 ∪ 𝐻))) |
13 | | mreexexlem2d.8 |
. . . 4
⊢ (𝜑 → (𝐹 ∪ 𝐻) ∈ 𝐼) |
14 | 13 | adantr 485 |
. . 3
⊢ ((𝜑 ∧ 𝐹 = ∅) → (𝐹 ∪ 𝐻) ∈ 𝐼) |
15 | | animorrl 979 |
. . 3
⊢ ((𝜑 ∧ 𝐹 = ∅) → (𝐹 = ∅ ∨ 𝐺 = ∅)) |
16 | 2, 3, 4, 6, 8, 10,
12, 14, 15 | mreexexlem3d 16960 |
. 2
⊢ ((𝜑 ∧ 𝐹 = ∅) → ∃𝑗 ∈ 𝒫 𝐺(𝐹 ≈ 𝑗 ∧ (𝑗 ∪ 𝐻) ∈ 𝐼)) |
17 | | n0 4239 |
. . . . 5
⊢ (𝐹 ≠ ∅ ↔
∃𝑟 𝑟 ∈ 𝐹) |
18 | 17 | biimpi 219 |
. . . 4
⊢ (𝐹 ≠ ∅ →
∃𝑟 𝑟 ∈ 𝐹) |
19 | 18 | adantl 486 |
. . 3
⊢ ((𝜑 ∧ 𝐹 ≠ ∅) → ∃𝑟 𝑟 ∈ 𝐹) |
20 | 1 | adantr 485 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑟 ∈ 𝐹) → 𝐴 ∈ (Moore‘𝑋)) |
21 | 5 | adantr 485 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑟 ∈ 𝐹) → ∀𝑠 ∈ 𝒫 𝑋∀𝑦 ∈ 𝑋 ∀𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁‘𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧}))) |
22 | 7 | adantr 485 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑟 ∈ 𝐹) → 𝐹 ⊆ (𝑋 ∖ 𝐻)) |
23 | 9 | adantr 485 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑟 ∈ 𝐹) → 𝐺 ⊆ (𝑋 ∖ 𝐻)) |
24 | 11 | adantr 485 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑟 ∈ 𝐹) → 𝐹 ⊆ (𝑁‘(𝐺 ∪ 𝐻))) |
25 | 13 | adantr 485 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑟 ∈ 𝐹) → (𝐹 ∪ 𝐻) ∈ 𝐼) |
26 | | simpr 489 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑟 ∈ 𝐹) → 𝑟 ∈ 𝐹) |
27 | 20, 3, 4, 21, 22, 23, 24, 25, 26 | mreexexlem2d 16959 |
. . . . 5
⊢ ((𝜑 ∧ 𝑟 ∈ 𝐹) → ∃𝑞 ∈ 𝐺 (¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) |
28 | | 3anass 1093 |
. . . . . 6
⊢ ((𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼) ↔ (𝑞 ∈ 𝐺 ∧ (¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) |
29 | 1 | ad2antrr 726 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → 𝐴 ∈ (Moore‘𝑋)) |
30 | 29 | elfvexd 6685 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → 𝑋 ∈ V) |
31 | | simpr2 1193 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → ¬ 𝑞 ∈ (𝐹 ∖ {𝑟})) |
32 | | difsnb 4689 |
. . . . . . . . . . 11
⊢ (¬
𝑞 ∈ (𝐹 ∖ {𝑟}) ↔ ((𝐹 ∖ {𝑟}) ∖ {𝑞}) = (𝐹 ∖ {𝑟})) |
33 | 31, 32 | sylib 221 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → ((𝐹 ∖ {𝑟}) ∖ {𝑞}) = (𝐹 ∖ {𝑟})) |
34 | 7 | ad2antrr 726 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → 𝐹 ⊆ (𝑋 ∖ 𝐻)) |
35 | 34 | ssdifssd 4044 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → (𝐹 ∖ {𝑟}) ⊆ (𝑋 ∖ 𝐻)) |
36 | 35 | ssdifd 4042 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → ((𝐹 ∖ {𝑟}) ∖ {𝑞}) ⊆ ((𝑋 ∖ 𝐻) ∖ {𝑞})) |
37 | 33, 36 | eqsstrrd 3927 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → (𝐹 ∖ {𝑟}) ⊆ ((𝑋 ∖ 𝐻) ∖ {𝑞})) |
38 | | difun1 4190 |
. . . . . . . . 9
⊢ (𝑋 ∖ (𝐻 ∪ {𝑞})) = ((𝑋 ∖ 𝐻) ∖ {𝑞}) |
39 | 37, 38 | sseqtrrdi 3939 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → (𝐹 ∖ {𝑟}) ⊆ (𝑋 ∖ (𝐻 ∪ {𝑞}))) |
40 | 9 | ad2antrr 726 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → 𝐺 ⊆ (𝑋 ∖ 𝐻)) |
41 | 40 | ssdifd 4042 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → (𝐺 ∖ {𝑞}) ⊆ ((𝑋 ∖ 𝐻) ∖ {𝑞})) |
42 | 41, 38 | sseqtrrdi 3939 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → (𝐺 ∖ {𝑞}) ⊆ (𝑋 ∖ (𝐻 ∪ {𝑞}))) |
43 | 11 | ad2antrr 726 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → 𝐹 ⊆ (𝑁‘(𝐺 ∪ 𝐻))) |
44 | | simpr1 1192 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → 𝑞 ∈ 𝐺) |
45 | | uncom 4054 |
. . . . . . . . . . . . . 14
⊢ (𝐻 ∪ {𝑞}) = ({𝑞} ∪ 𝐻) |
46 | 45 | uneq2i 4061 |
. . . . . . . . . . . . 13
⊢ ((𝐺 ∖ {𝑞}) ∪ (𝐻 ∪ {𝑞})) = ((𝐺 ∖ {𝑞}) ∪ ({𝑞} ∪ 𝐻)) |
47 | | unass 4067 |
. . . . . . . . . . . . . 14
⊢ (((𝐺 ∖ {𝑞}) ∪ {𝑞}) ∪ 𝐻) = ((𝐺 ∖ {𝑞}) ∪ ({𝑞} ∪ 𝐻)) |
48 | | difsnid 4693 |
. . . . . . . . . . . . . . 15
⊢ (𝑞 ∈ 𝐺 → ((𝐺 ∖ {𝑞}) ∪ {𝑞}) = 𝐺) |
49 | 48 | uneq1d 4063 |
. . . . . . . . . . . . . 14
⊢ (𝑞 ∈ 𝐺 → (((𝐺 ∖ {𝑞}) ∪ {𝑞}) ∪ 𝐻) = (𝐺 ∪ 𝐻)) |
50 | 47, 49 | syl5eqr 2808 |
. . . . . . . . . . . . 13
⊢ (𝑞 ∈ 𝐺 → ((𝐺 ∖ {𝑞}) ∪ ({𝑞} ∪ 𝐻)) = (𝐺 ∪ 𝐻)) |
51 | 46, 50 | syl5eq 2806 |
. . . . . . . . . . . 12
⊢ (𝑞 ∈ 𝐺 → ((𝐺 ∖ {𝑞}) ∪ (𝐻 ∪ {𝑞})) = (𝐺 ∪ 𝐻)) |
52 | 44, 51 | syl 17 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → ((𝐺 ∖ {𝑞}) ∪ (𝐻 ∪ {𝑞})) = (𝐺 ∪ 𝐻)) |
53 | 52 | fveq2d 6655 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → (𝑁‘((𝐺 ∖ {𝑞}) ∪ (𝐻 ∪ {𝑞}))) = (𝑁‘(𝐺 ∪ 𝐻))) |
54 | 43, 53 | sseqtrrd 3929 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → 𝐹 ⊆ (𝑁‘((𝐺 ∖ {𝑞}) ∪ (𝐻 ∪ {𝑞})))) |
55 | 54 | ssdifssd 4044 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → (𝐹 ∖ {𝑟}) ⊆ (𝑁‘((𝐺 ∖ {𝑞}) ∪ (𝐻 ∪ {𝑞})))) |
56 | | simpr3 1194 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼) |
57 | | mreexexlem4d.B |
. . . . . . . . . 10
⊢ (𝜑 → (𝐹 ≈ suc 𝐿 ∨ 𝐺 ≈ suc 𝐿)) |
58 | 57 | ad2antrr 726 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → (𝐹 ≈ suc 𝐿 ∨ 𝐺 ≈ suc 𝐿)) |
59 | | mreexexlem4d.9 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐿 ∈ ω) |
60 | 59 | ad2antrr 726 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → 𝐿 ∈ ω) |
61 | | simplr 769 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → 𝑟 ∈ 𝐹) |
62 | | 3anan12 1094 |
. . . . . . . . . . . . 13
⊢ ((𝐿 ∈ ω ∧ 𝐹 ≈ suc 𝐿 ∧ 𝑟 ∈ 𝐹) ↔ (𝐹 ≈ suc 𝐿 ∧ (𝐿 ∈ ω ∧ 𝑟 ∈ 𝐹))) |
63 | | dif1en 8767 |
. . . . . . . . . . . . 13
⊢ ((𝐿 ∈ ω ∧ 𝐹 ≈ suc 𝐿 ∧ 𝑟 ∈ 𝐹) → (𝐹 ∖ {𝑟}) ≈ 𝐿) |
64 | 62, 63 | sylbir 238 |
. . . . . . . . . . . 12
⊢ ((𝐹 ≈ suc 𝐿 ∧ (𝐿 ∈ ω ∧ 𝑟 ∈ 𝐹)) → (𝐹 ∖ {𝑟}) ≈ 𝐿) |
65 | 64 | expcom 418 |
. . . . . . . . . . 11
⊢ ((𝐿 ∈ ω ∧ 𝑟 ∈ 𝐹) → (𝐹 ≈ suc 𝐿 → (𝐹 ∖ {𝑟}) ≈ 𝐿)) |
66 | 60, 61, 65 | syl2anc 588 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → (𝐹 ≈ suc 𝐿 → (𝐹 ∖ {𝑟}) ≈ 𝐿)) |
67 | | 3anan12 1094 |
. . . . . . . . . . . . 13
⊢ ((𝐿 ∈ ω ∧ 𝐺 ≈ suc 𝐿 ∧ 𝑞 ∈ 𝐺) ↔ (𝐺 ≈ suc 𝐿 ∧ (𝐿 ∈ ω ∧ 𝑞 ∈ 𝐺))) |
68 | | dif1en 8767 |
. . . . . . . . . . . . 13
⊢ ((𝐿 ∈ ω ∧ 𝐺 ≈ suc 𝐿 ∧ 𝑞 ∈ 𝐺) → (𝐺 ∖ {𝑞}) ≈ 𝐿) |
69 | 67, 68 | sylbir 238 |
. . . . . . . . . . . 12
⊢ ((𝐺 ≈ suc 𝐿 ∧ (𝐿 ∈ ω ∧ 𝑞 ∈ 𝐺)) → (𝐺 ∖ {𝑞}) ≈ 𝐿) |
70 | 69 | expcom 418 |
. . . . . . . . . . 11
⊢ ((𝐿 ∈ ω ∧ 𝑞 ∈ 𝐺) → (𝐺 ≈ suc 𝐿 → (𝐺 ∖ {𝑞}) ≈ 𝐿)) |
71 | 60, 44, 70 | syl2anc 588 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → (𝐺 ≈ suc 𝐿 → (𝐺 ∖ {𝑞}) ≈ 𝐿)) |
72 | 66, 71 | orim12d 963 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → ((𝐹 ≈ suc 𝐿 ∨ 𝐺 ≈ suc 𝐿) → ((𝐹 ∖ {𝑟}) ≈ 𝐿 ∨ (𝐺 ∖ {𝑞}) ≈ 𝐿))) |
73 | 58, 72 | mpd 15 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → ((𝐹 ∖ {𝑟}) ≈ 𝐿 ∨ (𝐺 ∖ {𝑞}) ≈ 𝐿)) |
74 | | mreexexlem4d.A |
. . . . . . . . 9
⊢ (𝜑 → ∀ℎ∀𝑓 ∈ 𝒫 (𝑋 ∖ ℎ)∀𝑔 ∈ 𝒫 (𝑋 ∖ ℎ)(((𝑓 ≈ 𝐿 ∨ 𝑔 ≈ 𝐿) ∧ 𝑓 ⊆ (𝑁‘(𝑔 ∪ ℎ)) ∧ (𝑓 ∪ ℎ) ∈ 𝐼) → ∃𝑗 ∈ 𝒫 𝑔(𝑓 ≈ 𝑗 ∧ (𝑗 ∪ ℎ) ∈ 𝐼))) |
75 | 74 | ad2antrr 726 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → ∀ℎ∀𝑓 ∈ 𝒫 (𝑋 ∖ ℎ)∀𝑔 ∈ 𝒫 (𝑋 ∖ ℎ)(((𝑓 ≈ 𝐿 ∨ 𝑔 ≈ 𝐿) ∧ 𝑓 ⊆ (𝑁‘(𝑔 ∪ ℎ)) ∧ (𝑓 ∪ ℎ) ∈ 𝐼) → ∃𝑗 ∈ 𝒫 𝑔(𝑓 ≈ 𝑗 ∧ (𝑗 ∪ ℎ) ∈ 𝐼))) |
76 | 30, 39, 42, 55, 56, 73, 75 | mreexexlemd 16958 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → ∃𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞})((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) |
77 | 30 | adantr 485 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → 𝑋 ∈ V) |
78 | 9 | ad3antrrr 730 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → 𝐺 ⊆ (𝑋 ∖ 𝐻)) |
79 | 78 | difss2d 4036 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → 𝐺 ⊆ 𝑋) |
80 | 77, 79 | ssexd 5187 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → 𝐺 ∈ V) |
81 | | simprl 771 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → 𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞})) |
82 | 81 | elpwid 4498 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → 𝑖 ⊆ (𝐺 ∖ {𝑞})) |
83 | 82 | difss2d 4036 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → 𝑖 ⊆ 𝐺) |
84 | | simplr1 1213 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → 𝑞 ∈ 𝐺) |
85 | 84 | snssd 4692 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → {𝑞} ⊆ 𝐺) |
86 | 83, 85 | unssd 4087 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → (𝑖 ∪ {𝑞}) ⊆ 𝐺) |
87 | 80, 86 | sselpwd 5189 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → (𝑖 ∪ {𝑞}) ∈ 𝒫 𝐺) |
88 | | difsnid 4693 |
. . . . . . . . . 10
⊢ (𝑟 ∈ 𝐹 → ((𝐹 ∖ {𝑟}) ∪ {𝑟}) = 𝐹) |
89 | 88 | ad3antlr 731 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → ((𝐹 ∖ {𝑟}) ∪ {𝑟}) = 𝐹) |
90 | | simprrl 781 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → (𝐹 ∖ {𝑟}) ≈ 𝑖) |
91 | | en2sn 8605 |
. . . . . . . . . . . 12
⊢ ((𝑟 ∈ V ∧ 𝑞 ∈ V) → {𝑟} ≈ {𝑞}) |
92 | 91 | el2v 3415 |
. . . . . . . . . . 11
⊢ {𝑟} ≈ {𝑞} |
93 | 92 | a1i 11 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → {𝑟} ≈ {𝑞}) |
94 | | incom 4102 |
. . . . . . . . . . . 12
⊢ ((𝐹 ∖ {𝑟}) ∩ {𝑟}) = ({𝑟} ∩ (𝐹 ∖ {𝑟})) |
95 | | disjdif 4361 |
. . . . . . . . . . . 12
⊢ ({𝑟} ∩ (𝐹 ∖ {𝑟})) = ∅ |
96 | 94, 95 | eqtri 2782 |
. . . . . . . . . . 11
⊢ ((𝐹 ∖ {𝑟}) ∩ {𝑟}) = ∅ |
97 | 96 | a1i 11 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → ((𝐹 ∖ {𝑟}) ∩ {𝑟}) = ∅) |
98 | | ssdifin0 4372 |
. . . . . . . . . . 11
⊢ (𝑖 ⊆ (𝐺 ∖ {𝑞}) → (𝑖 ∩ {𝑞}) = ∅) |
99 | 82, 98 | syl 17 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → (𝑖 ∩ {𝑞}) = ∅) |
100 | | unen 8609 |
. . . . . . . . . 10
⊢ ((((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ {𝑟} ≈ {𝑞}) ∧ (((𝐹 ∖ {𝑟}) ∩ {𝑟}) = ∅ ∧ (𝑖 ∩ {𝑞}) = ∅)) → ((𝐹 ∖ {𝑟}) ∪ {𝑟}) ≈ (𝑖 ∪ {𝑞})) |
101 | 90, 93, 97, 99, 100 | syl22anc 838 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → ((𝐹 ∖ {𝑟}) ∪ {𝑟}) ≈ (𝑖 ∪ {𝑞})) |
102 | 89, 101 | eqbrtrrd 5049 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → 𝐹 ≈ (𝑖 ∪ {𝑞})) |
103 | | unass 4067 |
. . . . . . . . . 10
⊢ ((𝑖 ∪ {𝑞}) ∪ 𝐻) = (𝑖 ∪ ({𝑞} ∪ 𝐻)) |
104 | | uncom 4054 |
. . . . . . . . . . 11
⊢ ({𝑞} ∪ 𝐻) = (𝐻 ∪ {𝑞}) |
105 | 104 | uneq2i 4061 |
. . . . . . . . . 10
⊢ (𝑖 ∪ ({𝑞} ∪ 𝐻)) = (𝑖 ∪ (𝐻 ∪ {𝑞})) |
106 | 103, 105 | eqtr2i 2783 |
. . . . . . . . 9
⊢ (𝑖 ∪ (𝐻 ∪ {𝑞})) = ((𝑖 ∪ {𝑞}) ∪ 𝐻) |
107 | | simprrr 782 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼) |
108 | 106, 107 | eqeltrrid 2856 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → ((𝑖 ∪ {𝑞}) ∪ 𝐻) ∈ 𝐼) |
109 | | breq2 5029 |
. . . . . . . . . 10
⊢ (𝑗 = (𝑖 ∪ {𝑞}) → (𝐹 ≈ 𝑗 ↔ 𝐹 ≈ (𝑖 ∪ {𝑞}))) |
110 | | uneq1 4057 |
. . . . . . . . . . 11
⊢ (𝑗 = (𝑖 ∪ {𝑞}) → (𝑗 ∪ 𝐻) = ((𝑖 ∪ {𝑞}) ∪ 𝐻)) |
111 | 110 | eleq1d 2835 |
. . . . . . . . . 10
⊢ (𝑗 = (𝑖 ∪ {𝑞}) → ((𝑗 ∪ 𝐻) ∈ 𝐼 ↔ ((𝑖 ∪ {𝑞}) ∪ 𝐻) ∈ 𝐼)) |
112 | 109, 111 | anbi12d 634 |
. . . . . . . . 9
⊢ (𝑗 = (𝑖 ∪ {𝑞}) → ((𝐹 ≈ 𝑗 ∧ (𝑗 ∪ 𝐻) ∈ 𝐼) ↔ (𝐹 ≈ (𝑖 ∪ {𝑞}) ∧ ((𝑖 ∪ {𝑞}) ∪ 𝐻) ∈ 𝐼))) |
113 | 112 | rspcev 3539 |
. . . . . . . 8
⊢ (((𝑖 ∪ {𝑞}) ∈ 𝒫 𝐺 ∧ (𝐹 ≈ (𝑖 ∪ {𝑞}) ∧ ((𝑖 ∪ {𝑞}) ∪ 𝐻) ∈ 𝐼)) → ∃𝑗 ∈ 𝒫 𝐺(𝐹 ≈ 𝑗 ∧ (𝑗 ∪ 𝐻) ∈ 𝐼)) |
114 | 87, 102, 108, 113 | syl12anc 836 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) ∧ (𝑖 ∈ 𝒫 (𝐺 ∖ {𝑞}) ∧ ((𝐹 ∖ {𝑟}) ≈ 𝑖 ∧ (𝑖 ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → ∃𝑗 ∈ 𝒫 𝐺(𝐹 ≈ 𝑗 ∧ (𝑗 ∪ 𝐻) ∈ 𝐼)) |
115 | 76, 114 | rexlimddv 3213 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ ¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼)) → ∃𝑗 ∈ 𝒫 𝐺(𝐹 ≈ 𝑗 ∧ (𝑗 ∪ 𝐻) ∈ 𝐼)) |
116 | 28, 115 | sylan2br 598 |
. . . . 5
⊢ (((𝜑 ∧ 𝑟 ∈ 𝐹) ∧ (𝑞 ∈ 𝐺 ∧ (¬ 𝑞 ∈ (𝐹 ∖ {𝑟}) ∧ ((𝐹 ∖ {𝑟}) ∪ (𝐻 ∪ {𝑞})) ∈ 𝐼))) → ∃𝑗 ∈ 𝒫 𝐺(𝐹 ≈ 𝑗 ∧ (𝑗 ∪ 𝐻) ∈ 𝐼)) |
117 | 27, 116 | rexlimddv 3213 |
. . . 4
⊢ ((𝜑 ∧ 𝑟 ∈ 𝐹) → ∃𝑗 ∈ 𝒫 𝐺(𝐹 ≈ 𝑗 ∧ (𝑗 ∪ 𝐻) ∈ 𝐼)) |
118 | 117 | adantlr 715 |
. . 3
⊢ (((𝜑 ∧ 𝐹 ≠ ∅) ∧ 𝑟 ∈ 𝐹) → ∃𝑗 ∈ 𝒫 𝐺(𝐹 ≈ 𝑗 ∧ (𝑗 ∪ 𝐻) ∈ 𝐼)) |
119 | 19, 118 | exlimddv 1937 |
. 2
⊢ ((𝜑 ∧ 𝐹 ≠ ∅) → ∃𝑗 ∈ 𝒫 𝐺(𝐹 ≈ 𝑗 ∧ (𝑗 ∪ 𝐻) ∈ 𝐼)) |
120 | 16, 119 | pm2.61dane 3036 |
1
⊢ (𝜑 → ∃𝑗 ∈ 𝒫 𝐺(𝐹 ≈ 𝑗 ∧ (𝑗 ∪ 𝐻) ∈ 𝐼)) |