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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 7386. (Contributed by NM, 14-Sep-2003.) |
Ref | Expression |
---|---|
pwv | ⊢ 𝒫 V = V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssv 4020 | . . . 4 ⊢ 𝑥 ⊆ V | |
2 | velpw 4610 | . . . 4 ⊢ (𝑥 ∈ 𝒫 V ↔ 𝑥 ⊆ V) | |
3 | 1, 2 | mpbir 231 | . . 3 ⊢ 𝑥 ∈ 𝒫 V |
4 | vex 3482 | . . 3 ⊢ 𝑥 ∈ V | |
5 | 3, 4 | 2th 264 | . 2 ⊢ (𝑥 ∈ 𝒫 V ↔ 𝑥 ∈ V) |
6 | 5 | eqriv 2732 | 1 ⊢ 𝒫 V = V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2106 Vcvv 3478 ⊆ wss 3963 𝒫 cpw 4605 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-ss 3980 df-pw 4607 |
This theorem is referenced by: univ 5462 ncanth 7386 |
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