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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. (Contributed by NM, 14-Sep-2003.) |
Ref | Expression |
---|---|
pwv | ⊢ 𝒫 V = V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssv 3907 | . . . 4 ⊢ 𝑥 ⊆ V | |
2 | selpw 4454 | . . . 4 ⊢ (𝑥 ∈ 𝒫 V ↔ 𝑥 ⊆ V) | |
3 | 1, 2 | mpbir 232 | . . 3 ⊢ 𝑥 ∈ 𝒫 V |
4 | vex 3435 | . . 3 ⊢ 𝑥 ∈ V | |
5 | 3, 4 | 2th 265 | . 2 ⊢ (𝑥 ∈ 𝒫 V ↔ 𝑥 ∈ V) |
6 | 5 | eqriv 2790 | 1 ⊢ 𝒫 V = V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1520 ∈ wcel 2079 Vcvv 3432 ⊆ wss 3854 𝒫 cpw 4447 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-ext 2767 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-v 3434 df-in 3861 df-ss 3869 df-pw 4449 |
This theorem is referenced by: univ 5228 ncanth 6966 |
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