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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 3625 | . . . 4 ⊢ 𝑥 ⊆ V | |
2 | selpw 4165 | . . . 4 ⊢ (𝑥 ∈ 𝒫 V ↔ 𝑥 ⊆ V) | |
3 | 1, 2 | mpbir 221 | . . 3 ⊢ 𝑥 ∈ 𝒫 V |
4 | vex 3203 | . . 3 ⊢ 𝑥 ∈ V | |
5 | 3, 4 | 2th 254 | . 2 ⊢ (𝑥 ∈ 𝒫 V ↔ 𝑥 ∈ V) |
6 | 5 | eqriv 2619 | 1 ⊢ 𝒫 V = V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ∈ wcel 1990 Vcvv 3200 ⊆ wss 3574 𝒫 cpw 4158 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-in 3581 df-ss 3588 df-pw 4160 |
This theorem is referenced by: univ 4919 ncanth 6609 |
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