| Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | ||
| Mirrors > Home > MPE Home > Th. List > pwv | Structured version Visualization version GIF version | ||
| Description: The power class of the
universe is the universe.  Exercise 4.12(d) of
     [Mendelson] p. 235. The collection of all classes is of course larger than V, which is the collection of all sets. But 𝒫 V, being a class, cannot contain proper classes, so 𝒫 V is actually no larger than V. This fact is exploited in ncanth 7387. (Contributed by NM, 14-Sep-2003.) | 
| Ref | Expression | 
|---|---|
| pwv | ⊢ 𝒫 V = V | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ssv 4007 | . . . 4 ⊢ 𝑥 ⊆ V | |
| 2 | velpw 4604 | . . . 4 ⊢ (𝑥 ∈ 𝒫 V ↔ 𝑥 ⊆ V) | |
| 3 | 1, 2 | mpbir 231 | . . 3 ⊢ 𝑥 ∈ 𝒫 V | 
| 4 | vex 3483 | . . 3 ⊢ 𝑥 ∈ V | |
| 5 | 3, 4 | 2th 264 | . 2 ⊢ (𝑥 ∈ 𝒫 V ↔ 𝑥 ∈ V) | 
| 6 | 5 | eqriv 2733 | 1 ⊢ 𝒫 V = V | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 ∈ wcel 2107 Vcvv 3479 ⊆ wss 3950 𝒫 cpw 4599 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-v 3481 df-ss 3967 df-pw 4601 | 
| This theorem is referenced by: univ 5455 ncanth 7387 | 
| Copyright terms: Public domain | W3C validator |