Proof of Theorem ex-sategoelel
Step | Hyp | Ref
| Expression |
1 | | simpr 488 |
. . . . . . 7
⊢ ((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) → 𝑍 ∈ 𝑀) |
2 | | simpl 486 |
. . . . . . . . 9
⊢ ((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) → 𝑀 ∈ WUni) |
3 | 2, 1 | wunpw 10321 |
. . . . . . . 8
⊢ ((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) → 𝒫 𝑍 ∈ 𝑀) |
4 | 2 | wun0 10332 |
. . . . . . . 8
⊢ ((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) → ∅ ∈ 𝑀) |
5 | 3, 4 | ifcld 4485 |
. . . . . . 7
⊢ ((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) → if(𝑥 = 𝐵, 𝒫 𝑍, ∅) ∈ 𝑀) |
6 | 1, 5 | ifcld 4485 |
. . . . . 6
⊢ ((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) → if(𝑥 = 𝐴, 𝑍, if(𝑥 = 𝐵, 𝒫 𝑍, ∅)) ∈ 𝑀) |
7 | 6 | adantr 484 |
. . . . 5
⊢ (((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ≠ 𝐵)) → if(𝑥 = 𝐴, 𝑍, if(𝑥 = 𝐵, 𝒫 𝑍, ∅)) ∈ 𝑀) |
8 | 7 | adantr 484 |
. . . 4
⊢ ((((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ≠ 𝐵)) ∧ 𝑥 ∈ ω) → if(𝑥 = 𝐴, 𝑍, if(𝑥 = 𝐵, 𝒫 𝑍, ∅)) ∈ 𝑀) |
9 | | ex-sategoelel.s |
. . . 4
⊢ 𝑆 = (𝑥 ∈ ω ↦ if(𝑥 = 𝐴, 𝑍, if(𝑥 = 𝐵, 𝒫 𝑍, ∅))) |
10 | 8, 9 | fmptd 6931 |
. . 3
⊢ (((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ≠ 𝐵)) → 𝑆:ω⟶𝑀) |
11 | 2 | adantr 484 |
. . . 4
⊢ (((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ≠ 𝐵)) → 𝑀 ∈ WUni) |
12 | | omex 9258 |
. . . . 5
⊢ ω
∈ V |
13 | 12 | a1i 11 |
. . . 4
⊢ (((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ≠ 𝐵)) → ω ∈ V) |
14 | 11, 13 | elmapd 8522 |
. . 3
⊢ (((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ≠ 𝐵)) → (𝑆 ∈ (𝑀 ↑m ω) ↔ 𝑆:ω⟶𝑀)) |
15 | 10, 14 | mpbird 260 |
. 2
⊢ (((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ≠ 𝐵)) → 𝑆 ∈ (𝑀 ↑m
ω)) |
16 | | pwidg 4535 |
. . . . 5
⊢ (𝑍 ∈ 𝑀 → 𝑍 ∈ 𝒫 𝑍) |
17 | 16 | adantl 485 |
. . . 4
⊢ ((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) → 𝑍 ∈ 𝒫 𝑍) |
18 | 17 | adantr 484 |
. . 3
⊢ (((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ≠ 𝐵)) → 𝑍 ∈ 𝒫 𝑍) |
19 | 9 | a1i 11 |
. . . 4
⊢ (((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ≠ 𝐵)) → 𝑆 = (𝑥 ∈ ω ↦ if(𝑥 = 𝐴, 𝑍, if(𝑥 = 𝐵, 𝒫 𝑍, ∅)))) |
20 | | iftrue 4445 |
. . . . 5
⊢ (𝑥 = 𝐴 → if(𝑥 = 𝐴, 𝑍, if(𝑥 = 𝐵, 𝒫 𝑍, ∅)) = 𝑍) |
21 | 20 | adantl 485 |
. . . 4
⊢ ((((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ≠ 𝐵)) ∧ 𝑥 = 𝐴) → if(𝑥 = 𝐴, 𝑍, if(𝑥 = 𝐵, 𝒫 𝑍, ∅)) = 𝑍) |
22 | | simpr1 1196 |
. . . 4
⊢ (((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ≠ 𝐵)) → 𝐴 ∈ ω) |
23 | 1 | adantr 484 |
. . . 4
⊢ (((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ≠ 𝐵)) → 𝑍 ∈ 𝑀) |
24 | 19, 21, 22, 23 | fvmptd 6825 |
. . 3
⊢ (((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ≠ 𝐵)) → (𝑆‘𝐴) = 𝑍) |
25 | | eqeq1 2741 |
. . . . . . 7
⊢ (𝑥 = 𝐵 → (𝑥 = 𝐴 ↔ 𝐵 = 𝐴)) |
26 | | eqeq1 2741 |
. . . . . . . 8
⊢ (𝑥 = 𝐵 → (𝑥 = 𝐵 ↔ 𝐵 = 𝐵)) |
27 | 26 | ifbid 4462 |
. . . . . . 7
⊢ (𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝒫 𝑍, ∅) = if(𝐵 = 𝐵, 𝒫 𝑍, ∅)) |
28 | 25, 27 | ifbieq2d 4465 |
. . . . . 6
⊢ (𝑥 = 𝐵 → if(𝑥 = 𝐴, 𝑍, if(𝑥 = 𝐵, 𝒫 𝑍, ∅)) = if(𝐵 = 𝐴, 𝑍, if(𝐵 = 𝐵, 𝒫 𝑍, ∅))) |
29 | | necom 2994 |
. . . . . . . . 9
⊢ (𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴) |
30 | | ifnefalse 4451 |
. . . . . . . . 9
⊢ (𝐵 ≠ 𝐴 → if(𝐵 = 𝐴, 𝑍, if(𝐵 = 𝐵, 𝒫 𝑍, ∅)) = if(𝐵 = 𝐵, 𝒫 𝑍, ∅)) |
31 | 29, 30 | sylbi 220 |
. . . . . . . 8
⊢ (𝐴 ≠ 𝐵 → if(𝐵 = 𝐴, 𝑍, if(𝐵 = 𝐵, 𝒫 𝑍, ∅)) = if(𝐵 = 𝐵, 𝒫 𝑍, ∅)) |
32 | 31 | 3ad2ant3 1137 |
. . . . . . 7
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ≠ 𝐵) → if(𝐵 = 𝐴, 𝑍, if(𝐵 = 𝐵, 𝒫 𝑍, ∅)) = if(𝐵 = 𝐵, 𝒫 𝑍, ∅)) |
33 | 32 | adantl 485 |
. . . . . 6
⊢ (((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ≠ 𝐵)) → if(𝐵 = 𝐴, 𝑍, if(𝐵 = 𝐵, 𝒫 𝑍, ∅)) = if(𝐵 = 𝐵, 𝒫 𝑍, ∅)) |
34 | 28, 33 | sylan9eqr 2800 |
. . . . 5
⊢ ((((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ≠ 𝐵)) ∧ 𝑥 = 𝐵) → if(𝑥 = 𝐴, 𝑍, if(𝑥 = 𝐵, 𝒫 𝑍, ∅)) = if(𝐵 = 𝐵, 𝒫 𝑍, ∅)) |
35 | | simpr2 1197 |
. . . . 5
⊢ (((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ≠ 𝐵)) → 𝐵 ∈ ω) |
36 | | pwexg 5271 |
. . . . . . . 8
⊢ (𝑍 ∈ 𝑀 → 𝒫 𝑍 ∈ V) |
37 | 36 | adantl 485 |
. . . . . . 7
⊢ ((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) → 𝒫 𝑍 ∈ V) |
38 | | 0ex 5200 |
. . . . . . . 8
⊢ ∅
∈ V |
39 | 38 | a1i 11 |
. . . . . . 7
⊢ ((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) → ∅ ∈ V) |
40 | 37, 39 | ifcld 4485 |
. . . . . 6
⊢ ((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) → if(𝐵 = 𝐵, 𝒫 𝑍, ∅) ∈ V) |
41 | 40 | adantr 484 |
. . . . 5
⊢ (((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ≠ 𝐵)) → if(𝐵 = 𝐵, 𝒫 𝑍, ∅) ∈ V) |
42 | 19, 34, 35, 41 | fvmptd 6825 |
. . . 4
⊢ (((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ≠ 𝐵)) → (𝑆‘𝐵) = if(𝐵 = 𝐵, 𝒫 𝑍, ∅)) |
43 | | eqid 2737 |
. . . . 5
⊢ 𝐵 = 𝐵 |
44 | 43 | iftruei 4446 |
. . . 4
⊢ if(𝐵 = 𝐵, 𝒫 𝑍, ∅) = 𝒫 𝑍 |
45 | 42, 44 | eqtrdi 2794 |
. . 3
⊢ (((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ≠ 𝐵)) → (𝑆‘𝐵) = 𝒫 𝑍) |
46 | 18, 24, 45 | 3eltr4d 2853 |
. 2
⊢ (((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ≠ 𝐵)) → (𝑆‘𝐴) ∈ (𝑆‘𝐵)) |
47 | | 3simpa 1150 |
. . 3
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ≠ 𝐵) → (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) |
48 | | sategoelfvb.s |
. . . 4
⊢ 𝐸 = (𝑀 Sat∈ (𝐴∈𝑔𝐵)) |
49 | 48 | sategoelfvb 33094 |
. . 3
⊢ ((𝑀 ∈ WUni ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝑆 ∈ 𝐸 ↔ (𝑆 ∈ (𝑀 ↑m ω) ∧ (𝑆‘𝐴) ∈ (𝑆‘𝐵)))) |
50 | 2, 47, 49 | syl2an 599 |
. 2
⊢ (((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ≠ 𝐵)) → (𝑆 ∈ 𝐸 ↔ (𝑆 ∈ (𝑀 ↑m ω) ∧ (𝑆‘𝐴) ∈ (𝑆‘𝐵)))) |
51 | 15, 46, 50 | mpbir2and 713 |
1
⊢ (((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ≠ 𝐵)) → 𝑆 ∈ 𝐸) |