Theorem pwexb 4571

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.)

Assertion

Ref	Expression
pwexb	⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)

Proof of Theorem pwexb

Step	Hyp	Ref	Expression
1		uniexb 4570	. 2 ⊢ (𝒫 𝐴 ∈ V ↔ ∪ 𝒫 𝐴 ∈ V)
2		unipw 4309	. . 3 ⊢ ∪ 𝒫 𝐴 = 𝐴
3	2	eleq1i 2297	. 2 ⊢ (∪ 𝒫 𝐴 ∈ V ↔ 𝐴 ∈ V)
4	1, 3	bitr2i 185	1 ⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)

Syntax hints:   ↔ wb 105   ∈ wcel 2202  Vcvv 2802  𝒫 cpw 3652  ∪ cuni 3893

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-v 2804  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-uni 3894

This theorem is referenced by: (None)