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Mirrors > Home > ILE Home > Th. List > pwv | GIF version |
Description: The power class of the universe is the universe. Exercise 4.12(d) of [Mendelson] p. 235. (Contributed by NM, 14-Sep-2003.) |
Ref | Expression |
---|---|
pwv | ⊢ 𝒫 V = V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssv 3019 | . . . 4 ⊢ 𝑥 ⊆ V | |
2 | vex 2604 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | 2 | elpw 3388 | . . . 4 ⊢ (𝑥 ∈ 𝒫 V ↔ 𝑥 ⊆ V) |
4 | 1, 3 | mpbir 144 | . . 3 ⊢ 𝑥 ∈ 𝒫 V |
5 | 4, 2 | 2th 172 | . 2 ⊢ (𝑥 ∈ 𝒫 V ↔ 𝑥 ∈ V) |
6 | 5 | eqriv 2078 | 1 ⊢ 𝒫 V = V |
Colors of variables: wff set class |
Syntax hints: = wceq 1284 ∈ wcel 1433 Vcvv 2601 ⊆ wss 2973 𝒫 cpw 3382 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-in 2979 df-ss 2986 df-pw 3384 |
This theorem is referenced by: univ 4225 |
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