![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > pwexb | Structured version Visualization version GIF version |
Description: The Axiom of Power Sets and its converse. A class is a set iff its power class is a set. (Contributed by NM, 11-Nov-2003.) |
Ref | Expression |
---|---|
pwexb | ⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwexg 5382 | . 2 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ V) | |
2 | pwexr 7773 | . 2 ⊢ (𝒫 𝐴 ∈ V → 𝐴 ∈ V) | |
3 | 1, 2 | impbii 208 | 1 ⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∈ wcel 2098 Vcvv 3473 𝒫 cpw 4606 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2699 ax-sep 5303 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-v 3475 df-un 3954 df-in 3956 df-ss 3966 df-pw 4608 df-sn 4633 df-pr 4635 df-uni 4913 |
This theorem is referenced by: 2pwuninel 9163 ranklim 9875 r1pwALT 9877 isf34lem6 10411 isfin1-2 10416 pwfseqlem4 10693 pwfseqlem5 10694 gchpwdom 10701 hargch 10704 numufl 23839 |
Copyright terms: Public domain | W3C validator |