Theorem pwexb 4459

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 4458	. 2 ⊢ (𝒫 𝐴 ∈ V ↔ ∪ 𝒫 𝐴 ∈ V)
2		unipw 4202	. . 3 ⊢ ∪ 𝒫 𝐴 = 𝐴
3	2	eleq1i 2236	. 2 ⊢ (∪ 𝒫 𝐴 ∈ V ↔ 𝐴 ∈ V)
4	1, 3	bitr2i 184	1 ⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)

Syntax hints:   ↔ wb 104   ∈ wcel 2141  Vcvv 2730  𝒫 cpw 3566  ∪ cuni 3796

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-uni 3797

This theorem is referenced by: (None)