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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 3177 | . . . 4 ⊢ 𝑥 ⊆ V | |
2 | vex 2740 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | 2 | elpw 3580 | . . . 4 ⊢ (𝑥 ∈ 𝒫 V ↔ 𝑥 ⊆ V) |
4 | 1, 3 | mpbir 146 | . . 3 ⊢ 𝑥 ∈ 𝒫 V |
5 | 4, 2 | 2th 174 | . 2 ⊢ (𝑥 ∈ 𝒫 V ↔ 𝑥 ∈ V) |
6 | 5 | eqriv 2174 | 1 ⊢ 𝒫 V = V |
Colors of variables: wff set class |
Syntax hints: = wceq 1353 ∈ wcel 2148 Vcvv 2737 ⊆ wss 3129 𝒫 cpw 3574 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-in 3135 df-ss 3142 df-pw 3576 |
This theorem is referenced by: univ 4472 |
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