Proof of Theorem ex-sategoelel
| Step | Hyp | Ref
| Expression |
| 1 | | simpr 484 |
. . . . . . 7
⊢ ((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) → 𝑍 ∈ 𝑀) |
| 2 | | simpl 482 |
. . . . . . . . 9
⊢ ((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) → 𝑀 ∈ WUni) |
| 3 | 2, 1 | wunpw 10747 |
. . . . . . . 8
⊢ ((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) → 𝒫 𝑍 ∈ 𝑀) |
| 4 | 2 | wun0 10758 |
. . . . . . . 8
⊢ ((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) → ∅ ∈ 𝑀) |
| 5 | 3, 4 | ifcld 4572 |
. . . . . . 7
⊢ ((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) → if(𝑥 = 𝐵, 𝒫 𝑍, ∅) ∈ 𝑀) |
| 6 | 1, 5 | ifcld 4572 |
. . . . . 6
⊢ ((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) → if(𝑥 = 𝐴, 𝑍, if(𝑥 = 𝐵, 𝒫 𝑍, ∅)) ∈ 𝑀) |
| 7 | 6 | adantr 480 |
. . . . 5
⊢ (((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ≠ 𝐵)) → if(𝑥 = 𝐴, 𝑍, if(𝑥 = 𝐵, 𝒫 𝑍, ∅)) ∈ 𝑀) |
| 8 | 7 | adantr 480 |
. . . 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 7134 |
. . 3
⊢ (((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ≠ 𝐵)) → 𝑆:ω⟶𝑀) |
| 11 | 2 | adantr 480 |
. . . 4
⊢ (((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ≠ 𝐵)) → 𝑀 ∈ WUni) |
| 12 | | omex 9683 |
. . . . 5
⊢ ω
∈ V |
| 13 | 12 | a1i 11 |
. . . 4
⊢ (((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ≠ 𝐵)) → ω ∈ V) |
| 14 | 11, 13 | elmapd 8880 |
. . 3
⊢ (((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ≠ 𝐵)) → (𝑆 ∈ (𝑀 ↑m ω) ↔ 𝑆:ω⟶𝑀)) |
| 15 | 10, 14 | mpbird 257 |
. 2
⊢ (((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ≠ 𝐵)) → 𝑆 ∈ (𝑀 ↑m
ω)) |
| 16 | | pwidg 4620 |
. . . . 5
⊢ (𝑍 ∈ 𝑀 → 𝑍 ∈ 𝒫 𝑍) |
| 17 | 16 | adantl 481 |
. . . 4
⊢ ((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) → 𝑍 ∈ 𝒫 𝑍) |
| 18 | 17 | adantr 480 |
. . 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 4531 |
. . . . 5
⊢ (𝑥 = 𝐴 → if(𝑥 = 𝐴, 𝑍, if(𝑥 = 𝐵, 𝒫 𝑍, ∅)) = 𝑍) |
| 21 | 20 | adantl 481 |
. . . 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 1195 |
. . . 4
⊢ (((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ≠ 𝐵)) → 𝐴 ∈ ω) |
| 23 | 1 | adantr 480 |
. . . 4
⊢ (((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ≠ 𝐵)) → 𝑍 ∈ 𝑀) |
| 24 | 19, 21, 22, 23 | fvmptd 7023 |
. . 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 4549 |
. . . . . . 7
⊢ (𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝒫 𝑍, ∅) = if(𝐵 = 𝐵, 𝒫 𝑍, ∅)) |
| 28 | 25, 27 | ifbieq2d 4552 |
. . . . . 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 4537 |
. . . . . . . . 9
⊢ (𝐵 ≠ 𝐴 → if(𝐵 = 𝐴, 𝑍, if(𝐵 = 𝐵, 𝒫 𝑍, ∅)) = if(𝐵 = 𝐵, 𝒫 𝑍, ∅)) |
| 31 | 29, 30 | sylbi 217 |
. . . . . . . 8
⊢ (𝐴 ≠ 𝐵 → if(𝐵 = 𝐴, 𝑍, if(𝐵 = 𝐵, 𝒫 𝑍, ∅)) = if(𝐵 = 𝐵, 𝒫 𝑍, ∅)) |
| 32 | 31 | 3ad2ant3 1136 |
. . . . . . 7
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ≠ 𝐵) → if(𝐵 = 𝐴, 𝑍, if(𝐵 = 𝐵, 𝒫 𝑍, ∅)) = if(𝐵 = 𝐵, 𝒫 𝑍, ∅)) |
| 33 | 32 | adantl 481 |
. . . . . 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 2799 |
. . . . 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 1196 |
. . . . 5
⊢ (((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ≠ 𝐵)) → 𝐵 ∈ ω) |
| 36 | | pwexg 5378 |
. . . . . . . 8
⊢ (𝑍 ∈ 𝑀 → 𝒫 𝑍 ∈ V) |
| 37 | 36 | adantl 481 |
. . . . . . 7
⊢ ((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) → 𝒫 𝑍 ∈ V) |
| 38 | | 0ex 5307 |
. . . . . . . 8
⊢ ∅
∈ V |
| 39 | 38 | a1i 11 |
. . . . . . 7
⊢ ((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) → ∅ ∈ V) |
| 40 | 37, 39 | ifcld 4572 |
. . . . . 6
⊢ ((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) → if(𝐵 = 𝐵, 𝒫 𝑍, ∅) ∈ V) |
| 41 | 40 | adantr 480 |
. . . . 5
⊢ (((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ≠ 𝐵)) → if(𝐵 = 𝐵, 𝒫 𝑍, ∅) ∈ V) |
| 42 | 19, 34, 35, 41 | fvmptd 7023 |
. . . 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 4532 |
. . . 4
⊢ if(𝐵 = 𝐵, 𝒫 𝑍, ∅) = 𝒫 𝑍 |
| 45 | 42, 44 | eqtrdi 2793 |
. . 3
⊢ (((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ≠ 𝐵)) → (𝑆‘𝐵) = 𝒫 𝑍) |
| 46 | 18, 24, 45 | 3eltr4d 2856 |
. 2
⊢ (((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ≠ 𝐵)) → (𝑆‘𝐴) ∈ (𝑆‘𝐵)) |
| 47 | | 3simpa 1149 |
. . 3
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ≠ 𝐵) → (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) |
| 48 | | sategoelfvb.s |
. . . 4
⊢ 𝐸 = (𝑀 Sat∈ (𝐴∈𝑔𝐵)) |
| 49 | 48 | sategoelfvb 35424 |
. . 3
⊢ ((𝑀 ∈ WUni ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝑆 ∈ 𝐸 ↔ (𝑆 ∈ (𝑀 ↑m ω) ∧ (𝑆‘𝐴) ∈ (𝑆‘𝐵)))) |
| 50 | 2, 47, 49 | syl2an 596 |
. 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) |