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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 7312. (Contributed by NM, 14-Sep-2003.) |
Ref | Expression |
---|---|
pwv | ⊢ 𝒫 V = V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssv 3969 | . . . 4 ⊢ 𝑥 ⊆ V | |
2 | velpw 4566 | . . . 4 ⊢ (𝑥 ∈ 𝒫 V ↔ 𝑥 ⊆ V) | |
3 | 1, 2 | mpbir 230 | . . 3 ⊢ 𝑥 ∈ 𝒫 V |
4 | vex 3450 | . . 3 ⊢ 𝑥 ∈ V | |
5 | 3, 4 | 2th 264 | . 2 ⊢ (𝑥 ∈ 𝒫 V ↔ 𝑥 ∈ V) |
6 | 5 | eqriv 2734 | 1 ⊢ 𝒫 V = V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 Vcvv 3446 ⊆ wss 3911 𝒫 cpw 4561 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-v 3448 df-in 3918 df-ss 3928 df-pw 4563 |
This theorem is referenced by: univ 5409 ncanth 7312 |
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