Step | Hyp | Ref
| Expression |
1 | | simpr 477 |
. . 3
⊢ ((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or
𝐴 ∧ ¬ ∪ 𝐴
∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII)
→ 𝐵 ∈
FinIII) |
2 | | simpll1 1100 |
. . . . . . 7
⊢ ((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or
𝐴 ∧ ¬ ∪ 𝐴
∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII)
→ 𝐴 ⊆ 𝒫
𝐵) |
3 | 2 | adantr 481 |
. . . . . 6
⊢
(((((𝐴 ⊆
𝒫 𝐵 ∧
[⊊] Or 𝐴
∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) ∧ 𝑒 ∈ ω) → 𝐴 ⊆ 𝒫 𝐵) |
4 | | ssrab2 3687 |
. . . . . . . 8
⊢ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒} ⊆ 𝐴 |
5 | 4 | unissi 4461 |
. . . . . . 7
⊢ ∪ {𝑓
∈ 𝐴 ∣ 𝑓 ≼ 𝑒} ⊆ ∪ 𝐴 |
6 | | sspwuni 4611 |
. . . . . . . 8
⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴
⊆ 𝐵) |
7 | 6 | biimpi 206 |
. . . . . . 7
⊢ (𝐴 ⊆ 𝒫 𝐵 → ∪ 𝐴
⊆ 𝐵) |
8 | 5, 7 | syl5ss 3614 |
. . . . . 6
⊢ (𝐴 ⊆ 𝒫 𝐵 → ∪ {𝑓
∈ 𝐴 ∣ 𝑓 ≼ 𝑒} ⊆ 𝐵) |
9 | 3, 8 | syl 17 |
. . . . 5
⊢
(((((𝐴 ⊆
𝒫 𝐵 ∧
[⊊] Or 𝐴
∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) ∧ 𝑒 ∈ ω) → ∪ {𝑓
∈ 𝐴 ∣ 𝑓 ≼ 𝑒} ⊆ 𝐵) |
10 | | elpw2g 4827 |
. . . . . 6
⊢ (𝐵 ∈ FinIII →
(∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒} ∈ 𝒫 𝐵 ↔ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒} ⊆ 𝐵)) |
11 | 10 | ad2antlr 763 |
. . . . 5
⊢
(((((𝐴 ⊆
𝒫 𝐵 ∧
[⊊] Or 𝐴
∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) ∧ 𝑒 ∈ ω) → (∪ {𝑓
∈ 𝐴 ∣ 𝑓 ≼ 𝑒} ∈ 𝒫 𝐵 ↔ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒} ⊆ 𝐵)) |
12 | 9, 11 | mpbird 247 |
. . . 4
⊢
(((((𝐴 ⊆
𝒫 𝐵 ∧
[⊊] Or 𝐴
∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) ∧ 𝑒 ∈ ω) → ∪ {𝑓
∈ 𝐴 ∣ 𝑓 ≼ 𝑒} ∈ 𝒫 𝐵) |
13 | | eqid 2622 |
. . . 4
⊢ (𝑒 ∈ ω ↦ ∪ {𝑓
∈ 𝐴 ∣ 𝑓 ≼ 𝑒}) = (𝑒 ∈ ω ↦ ∪ {𝑓
∈ 𝐴 ∣ 𝑓 ≼ 𝑒}) |
14 | 12, 13 | fmptd 6385 |
. . 3
⊢ ((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or
𝐴 ∧ ¬ ∪ 𝐴
∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII)
→ (𝑒 ∈ ω
↦ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒}):ω⟶𝒫 𝐵) |
15 | | vex 3203 |
. . . . . . . . . . 11
⊢ 𝑑 ∈ V |
16 | 15 | sucex 7011 |
. . . . . . . . . 10
⊢ suc 𝑑 ∈ V |
17 | | sssucid 5802 |
. . . . . . . . . 10
⊢ 𝑑 ⊆ suc 𝑑 |
18 | | ssdomg 8001 |
. . . . . . . . . 10
⊢ (suc
𝑑 ∈ V → (𝑑 ⊆ suc 𝑑 → 𝑑 ≼ suc 𝑑)) |
19 | 16, 17, 18 | mp2 9 |
. . . . . . . . 9
⊢ 𝑑 ≼ suc 𝑑 |
20 | | domtr 8009 |
. . . . . . . . 9
⊢ ((𝑓 ≼ 𝑑 ∧ 𝑑 ≼ suc 𝑑) → 𝑓 ≼ suc 𝑑) |
21 | 19, 20 | mpan2 707 |
. . . . . . . 8
⊢ (𝑓 ≼ 𝑑 → 𝑓 ≼ suc 𝑑) |
22 | 21 | a1i 11 |
. . . . . . 7
⊢ (𝑓 ∈ 𝐴 → (𝑓 ≼ 𝑑 → 𝑓 ≼ suc 𝑑)) |
23 | 22 | ss2rabi 3684 |
. . . . . 6
⊢ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑑} ⊆ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ suc 𝑑} |
24 | | uniss 4458 |
. . . . . 6
⊢ ({𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑑} ⊆ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ suc 𝑑} → ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑑} ⊆ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ suc 𝑑}) |
25 | 23, 24 | mp1i 13 |
. . . . 5
⊢
(((((𝐴 ⊆
𝒫 𝐵 ∧
[⊊] Or 𝐴
∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) ∧ 𝑑 ∈ ω) → ∪ {𝑓
∈ 𝐴 ∣ 𝑓 ≼ 𝑑} ⊆ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ suc 𝑑}) |
26 | | id 22 |
. . . . . 6
⊢ (𝑑 ∈ ω → 𝑑 ∈
ω) |
27 | | pwexg 4850 |
. . . . . . . . 9
⊢ (𝐵 ∈ FinIII →
𝒫 𝐵 ∈
V) |
28 | 27 | adantl 482 |
. . . . . . . 8
⊢ ((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or
𝐴 ∧ ¬ ∪ 𝐴
∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII)
→ 𝒫 𝐵 ∈
V) |
29 | 28, 2 | ssexd 4805 |
. . . . . . 7
⊢ ((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or
𝐴 ∧ ¬ ∪ 𝐴
∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII)
→ 𝐴 ∈
V) |
30 | | rabexg 4812 |
. . . . . . 7
⊢ (𝐴 ∈ V → {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑑} ∈ V) |
31 | | uniexg 6955 |
. . . . . . 7
⊢ ({𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑑} ∈ V → ∪ {𝑓
∈ 𝐴 ∣ 𝑓 ≼ 𝑑} ∈ V) |
32 | 29, 30, 31 | 3syl 18 |
. . . . . 6
⊢ ((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or
𝐴 ∧ ¬ ∪ 𝐴
∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII)
→ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑑} ∈ V) |
33 | | breq2 4657 |
. . . . . . . . 9
⊢ (𝑒 = 𝑑 → (𝑓 ≼ 𝑒 ↔ 𝑓 ≼ 𝑑)) |
34 | 33 | rabbidv 3189 |
. . . . . . . 8
⊢ (𝑒 = 𝑑 → {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒} = {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑑}) |
35 | 34 | unieqd 4446 |
. . . . . . 7
⊢ (𝑒 = 𝑑 → ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒} = ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑑}) |
36 | 35, 13 | fvmptg 6280 |
. . . . . 6
⊢ ((𝑑 ∈ ω ∧ ∪ {𝑓
∈ 𝐴 ∣ 𝑓 ≼ 𝑑} ∈ V) → ((𝑒 ∈ ω ↦ ∪ {𝑓
∈ 𝐴 ∣ 𝑓 ≼ 𝑒})‘𝑑) = ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑑}) |
37 | 26, 32, 36 | syl2anr 495 |
. . . . 5
⊢
(((((𝐴 ⊆
𝒫 𝐵 ∧
[⊊] Or 𝐴
∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) ∧ 𝑑 ∈ ω) → ((𝑒 ∈ ω ↦ ∪ {𝑓
∈ 𝐴 ∣ 𝑓 ≼ 𝑒})‘𝑑) = ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑑}) |
38 | | peano2 7086 |
. . . . . 6
⊢ (𝑑 ∈ ω → suc 𝑑 ∈
ω) |
39 | | rabexg 4812 |
. . . . . . 7
⊢ (𝐴 ∈ V → {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ suc 𝑑} ∈ V) |
40 | | uniexg 6955 |
. . . . . . 7
⊢ ({𝑓 ∈ 𝐴 ∣ 𝑓 ≼ suc 𝑑} ∈ V → ∪ {𝑓
∈ 𝐴 ∣ 𝑓 ≼ suc 𝑑} ∈ V) |
41 | 29, 39, 40 | 3syl 18 |
. . . . . 6
⊢ ((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or
𝐴 ∧ ¬ ∪ 𝐴
∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII)
→ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ suc 𝑑} ∈ V) |
42 | | breq2 4657 |
. . . . . . . . 9
⊢ (𝑒 = suc 𝑑 → (𝑓 ≼ 𝑒 ↔ 𝑓 ≼ suc 𝑑)) |
43 | 42 | rabbidv 3189 |
. . . . . . . 8
⊢ (𝑒 = suc 𝑑 → {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒} = {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ suc 𝑑}) |
44 | 43 | unieqd 4446 |
. . . . . . 7
⊢ (𝑒 = suc 𝑑 → ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒} = ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ suc 𝑑}) |
45 | 44, 13 | fvmptg 6280 |
. . . . . 6
⊢ ((suc
𝑑 ∈ ω ∧
∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ suc 𝑑} ∈ V) → ((𝑒 ∈ ω ↦ ∪ {𝑓
∈ 𝐴 ∣ 𝑓 ≼ 𝑒})‘suc 𝑑) = ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ suc 𝑑}) |
46 | 38, 41, 45 | syl2anr 495 |
. . . . 5
⊢
(((((𝐴 ⊆
𝒫 𝐵 ∧
[⊊] Or 𝐴
∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) ∧ 𝑑 ∈ ω) → ((𝑒 ∈ ω ↦ ∪ {𝑓
∈ 𝐴 ∣ 𝑓 ≼ 𝑒})‘suc 𝑑) = ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ suc 𝑑}) |
47 | 25, 37, 46 | 3sstr4d 3648 |
. . . 4
⊢
(((((𝐴 ⊆
𝒫 𝐵 ∧
[⊊] Or 𝐴
∧ ¬ ∪ 𝐴 ∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII) ∧ 𝑑 ∈ ω) → ((𝑒 ∈ ω ↦ ∪ {𝑓
∈ 𝐴 ∣ 𝑓 ≼ 𝑒})‘𝑑) ⊆ ((𝑒 ∈ ω ↦ ∪ {𝑓
∈ 𝐴 ∣ 𝑓 ≼ 𝑒})‘suc 𝑑)) |
48 | 47 | ralrimiva 2966 |
. . 3
⊢ ((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or
𝐴 ∧ ¬ ∪ 𝐴
∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII)
→ ∀𝑑 ∈
ω ((𝑒 ∈ ω
↦ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒})‘𝑑) ⊆ ((𝑒 ∈ ω ↦ ∪ {𝑓
∈ 𝐴 ∣ 𝑓 ≼ 𝑒})‘suc 𝑑)) |
49 | | fin34i 9203 |
. . 3
⊢ ((𝐵 ∈ FinIII ∧
(𝑒 ∈ ω ↦
∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒}):ω⟶𝒫 𝐵 ∧ ∀𝑑 ∈ ω ((𝑒 ∈ ω ↦ ∪ {𝑓
∈ 𝐴 ∣ 𝑓 ≼ 𝑒})‘𝑑) ⊆ ((𝑒 ∈ ω ↦ ∪ {𝑓
∈ 𝐴 ∣ 𝑓 ≼ 𝑒})‘suc 𝑑)) → ∪ ran
(𝑒 ∈ ω ↦
∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒}) ∈ ran (𝑒 ∈ ω ↦ ∪ {𝑓
∈ 𝐴 ∣ 𝑓 ≼ 𝑒})) |
50 | 1, 14, 48, 49 | syl3anc 1326 |
. 2
⊢ ((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or
𝐴 ∧ ¬ ∪ 𝐴
∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII)
→ ∪ ran (𝑒 ∈ ω ↦ ∪ {𝑓
∈ 𝐴 ∣ 𝑓 ≼ 𝑒}) ∈ ran (𝑒 ∈ ω ↦ ∪ {𝑓
∈ 𝐴 ∣ 𝑓 ≼ 𝑒})) |
51 | | fin1a2lem11 9232 |
. . . . . 6
⊢ ((
[⊊] Or 𝐴
∧ 𝐴 ⊆ Fin) →
ran (𝑒 ∈ ω
↦ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒}) = (𝐴 ∪ {∅})) |
52 | 51 | adantrr 753 |
. . . . 5
⊢ ((
[⊊] Or 𝐴
∧ (𝐴 ⊆ Fin ∧
𝐴 ≠ ∅)) → ran
(𝑒 ∈ ω ↦
∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒}) = (𝐴 ∪ {∅})) |
53 | 52 | 3ad2antl2 1224 |
. . . 4
⊢ (((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or
𝐴 ∧ ¬ ∪ 𝐴
∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) → ran
(𝑒 ∈ ω ↦
∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒}) = (𝐴 ∪ {∅})) |
54 | 53 | adantr 481 |
. . 3
⊢ ((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or
𝐴 ∧ ¬ ∪ 𝐴
∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII)
→ ran (𝑒 ∈
ω ↦ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒}) = (𝐴 ∪ {∅})) |
55 | | simpll3 1102 |
. . . . . 6
⊢ ((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or
𝐴 ∧ ¬ ∪ 𝐴
∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII)
→ ¬ ∪ 𝐴 ∈ 𝐴) |
56 | | simplrr 801 |
. . . . . . 7
⊢ ((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or
𝐴 ∧ ¬ ∪ 𝐴
∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII)
→ 𝐴 ≠
∅) |
57 | | sspwuni 4611 |
. . . . . . . . . . 11
⊢ (𝐴 ⊆ 𝒫 ∅
↔ ∪ 𝐴 ⊆ ∅) |
58 | | ss0b 3973 |
. . . . . . . . . . 11
⊢ (∪ 𝐴
⊆ ∅ ↔ ∪ 𝐴 = ∅) |
59 | 57, 58 | bitri 264 |
. . . . . . . . . 10
⊢ (𝐴 ⊆ 𝒫 ∅
↔ ∪ 𝐴 = ∅) |
60 | | pw0 4343 |
. . . . . . . . . . . . 13
⊢ 𝒫
∅ = {∅} |
61 | 60 | sseq2i 3630 |
. . . . . . . . . . . 12
⊢ (𝐴 ⊆ 𝒫 ∅
↔ 𝐴 ⊆
{∅}) |
62 | | sssn 4358 |
. . . . . . . . . . . 12
⊢ (𝐴 ⊆ {∅} ↔ (𝐴 = ∅ ∨ 𝐴 = {∅})) |
63 | 61, 62 | bitri 264 |
. . . . . . . . . . 11
⊢ (𝐴 ⊆ 𝒫 ∅
↔ (𝐴 = ∅ ∨
𝐴 =
{∅})) |
64 | | df-ne 2795 |
. . . . . . . . . . . 12
⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅) |
65 | | 0ex 4790 |
. . . . . . . . . . . . . . . . 17
⊢ ∅
∈ V |
66 | 65 | unisn 4451 |
. . . . . . . . . . . . . . . 16
⊢ ∪ {∅} = ∅ |
67 | 65 | snid 4208 |
. . . . . . . . . . . . . . . 16
⊢ ∅
∈ {∅} |
68 | 66, 67 | eqeltri 2697 |
. . . . . . . . . . . . . . 15
⊢ ∪ {∅} ∈ {∅} |
69 | | unieq 4444 |
. . . . . . . . . . . . . . . 16
⊢ (𝐴 = {∅} → ∪ 𝐴 =
∪ {∅}) |
70 | | id 22 |
. . . . . . . . . . . . . . . 16
⊢ (𝐴 = {∅} → 𝐴 = {∅}) |
71 | 69, 70 | eleq12d 2695 |
. . . . . . . . . . . . . . 15
⊢ (𝐴 = {∅} → (∪ 𝐴
∈ 𝐴 ↔ ∪ {∅} ∈ {∅})) |
72 | 68, 71 | mpbiri 248 |
. . . . . . . . . . . . . 14
⊢ (𝐴 = {∅} → ∪ 𝐴
∈ 𝐴) |
73 | 72 | orim2i 540 |
. . . . . . . . . . . . 13
⊢ ((𝐴 = ∅ ∨ 𝐴 = {∅}) → (𝐴 = ∅ ∨ ∪ 𝐴
∈ 𝐴)) |
74 | 73 | ord 392 |
. . . . . . . . . . . 12
⊢ ((𝐴 = ∅ ∨ 𝐴 = {∅}) → (¬
𝐴 = ∅ → ∪ 𝐴
∈ 𝐴)) |
75 | 64, 74 | syl5bi 232 |
. . . . . . . . . . 11
⊢ ((𝐴 = ∅ ∨ 𝐴 = {∅}) → (𝐴 ≠ ∅ → ∪ 𝐴
∈ 𝐴)) |
76 | 63, 75 | sylbi 207 |
. . . . . . . . . 10
⊢ (𝐴 ⊆ 𝒫 ∅
→ (𝐴 ≠ ∅
→ ∪ 𝐴 ∈ 𝐴)) |
77 | 59, 76 | sylbir 225 |
. . . . . . . . 9
⊢ (∪ 𝐴 =
∅ → (𝐴 ≠
∅ → ∪ 𝐴 ∈ 𝐴)) |
78 | 77 | com12 32 |
. . . . . . . 8
⊢ (𝐴 ≠ ∅ → (∪ 𝐴 =
∅ → ∪ 𝐴 ∈ 𝐴)) |
79 | 78 | con3d 148 |
. . . . . . 7
⊢ (𝐴 ≠ ∅ → (¬
∪ 𝐴 ∈ 𝐴 → ¬ ∪
𝐴 =
∅)) |
80 | 56, 55, 79 | sylc 65 |
. . . . . 6
⊢ ((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or
𝐴 ∧ ¬ ∪ 𝐴
∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII)
→ ¬ ∪ 𝐴 = ∅) |
81 | | ioran 511 |
. . . . . 6
⊢ (¬
(∪ 𝐴 ∈ 𝐴 ∨ ∪ 𝐴 = ∅) ↔ (¬ ∪ 𝐴
∈ 𝐴 ∧ ¬ ∪ 𝐴 =
∅)) |
82 | 55, 80, 81 | sylanbrc 698 |
. . . . 5
⊢ ((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or
𝐴 ∧ ¬ ∪ 𝐴
∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII)
→ ¬ (∪ 𝐴 ∈ 𝐴 ∨ ∪ 𝐴 = ∅)) |
83 | | uniun 4456 |
. . . . . . . 8
⊢ ∪ (𝐴
∪ {∅}) = (∪ 𝐴 ∪ ∪
{∅}) |
84 | 66 | uneq2i 3764 |
. . . . . . . 8
⊢ (∪ 𝐴
∪ ∪ {∅}) = (∪
𝐴 ∪
∅) |
85 | | un0 3967 |
. . . . . . . 8
⊢ (∪ 𝐴
∪ ∅) = ∪ 𝐴 |
86 | 83, 84, 85 | 3eqtri 2648 |
. . . . . . 7
⊢ ∪ (𝐴
∪ {∅}) = ∪ 𝐴 |
87 | 86 | eleq1i 2692 |
. . . . . 6
⊢ (∪ (𝐴
∪ {∅}) ∈ (𝐴
∪ {∅}) ↔ ∪ 𝐴 ∈ (𝐴 ∪ {∅})) |
88 | | elun 3753 |
. . . . . 6
⊢ (∪ 𝐴
∈ (𝐴 ∪ {∅})
↔ (∪ 𝐴 ∈ 𝐴 ∨ ∪ 𝐴 ∈
{∅})) |
89 | 65 | elsn2 4211 |
. . . . . . 7
⊢ (∪ 𝐴
∈ {∅} ↔ ∪ 𝐴 = ∅) |
90 | 89 | orbi2i 541 |
. . . . . 6
⊢ ((∪ 𝐴
∈ 𝐴 ∨ ∪ 𝐴
∈ {∅}) ↔ (∪ 𝐴 ∈ 𝐴 ∨ ∪ 𝐴 = ∅)) |
91 | 87, 88, 90 | 3bitri 286 |
. . . . 5
⊢ (∪ (𝐴
∪ {∅}) ∈ (𝐴
∪ {∅}) ↔ (∪ 𝐴 ∈ 𝐴 ∨ ∪ 𝐴 = ∅)) |
92 | 82, 91 | sylnibr 319 |
. . . 4
⊢ ((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or
𝐴 ∧ ¬ ∪ 𝐴
∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII)
→ ¬ ∪ (𝐴 ∪ {∅}) ∈ (𝐴 ∪ {∅})) |
93 | | unieq 4444 |
. . . . . 6
⊢ (ran
(𝑒 ∈ ω ↦
∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒}) = (𝐴 ∪ {∅}) → ∪ ran (𝑒 ∈ ω ↦ ∪ {𝑓
∈ 𝐴 ∣ 𝑓 ≼ 𝑒}) = ∪ (𝐴 ∪
{∅})) |
94 | | id 22 |
. . . . . 6
⊢ (ran
(𝑒 ∈ ω ↦
∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒}) = (𝐴 ∪ {∅}) → ran (𝑒 ∈ ω ↦ ∪ {𝑓
∈ 𝐴 ∣ 𝑓 ≼ 𝑒}) = (𝐴 ∪ {∅})) |
95 | 93, 94 | eleq12d 2695 |
. . . . 5
⊢ (ran
(𝑒 ∈ ω ↦
∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒}) = (𝐴 ∪ {∅}) → (∪ ran (𝑒 ∈ ω ↦ ∪ {𝑓
∈ 𝐴 ∣ 𝑓 ≼ 𝑒}) ∈ ran (𝑒 ∈ ω ↦ ∪ {𝑓
∈ 𝐴 ∣ 𝑓 ≼ 𝑒}) ↔ ∪ (𝐴 ∪ {∅}) ∈ (𝐴 ∪
{∅}))) |
96 | 95 | notbid 308 |
. . . 4
⊢ (ran
(𝑒 ∈ ω ↦
∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒}) = (𝐴 ∪ {∅}) → (¬ ∪ ran (𝑒 ∈ ω ↦ ∪ {𝑓
∈ 𝐴 ∣ 𝑓 ≼ 𝑒}) ∈ ran (𝑒 ∈ ω ↦ ∪ {𝑓
∈ 𝐴 ∣ 𝑓 ≼ 𝑒}) ↔ ¬ ∪
(𝐴 ∪ {∅}) ∈
(𝐴 ∪
{∅}))) |
97 | 92, 96 | syl5ibrcom 237 |
. . 3
⊢ ((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or
𝐴 ∧ ¬ ∪ 𝐴
∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII)
→ (ran (𝑒 ∈
ω ↦ ∪ {𝑓 ∈ 𝐴 ∣ 𝑓 ≼ 𝑒}) = (𝐴 ∪ {∅}) → ¬ ∪ ran (𝑒 ∈ ω ↦ ∪ {𝑓
∈ 𝐴 ∣ 𝑓 ≼ 𝑒}) ∈ ran (𝑒 ∈ ω ↦ ∪ {𝑓
∈ 𝐴 ∣ 𝑓 ≼ 𝑒}))) |
98 | 54, 97 | mpd 15 |
. 2
⊢ ((((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or
𝐴 ∧ ¬ ∪ 𝐴
∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) ∧ 𝐵 ∈ FinIII)
→ ¬ ∪ ran (𝑒 ∈ ω ↦ ∪ {𝑓
∈ 𝐴 ∣ 𝑓 ≼ 𝑒}) ∈ ran (𝑒 ∈ ω ↦ ∪ {𝑓
∈ 𝐴 ∣ 𝑓 ≼ 𝑒})) |
99 | 50, 98 | pm2.65da 600 |
1
⊢ (((𝐴 ⊆ 𝒫 𝐵 ∧ [⊊] Or
𝐴 ∧ ¬ ∪ 𝐴
∈ 𝐴) ∧ (𝐴 ⊆ Fin ∧ 𝐴 ≠ ∅)) → ¬
𝐵 ∈
FinIII) |