Theorem pwexb 4476

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 4475	. 2 ⊢ (𝒫 𝐴 ∈ V ↔ ∪ 𝒫 𝐴 ∈ V)
2		unipw 4219	. . 3 ⊢ ∪ 𝒫 𝐴 = 𝐴
3	2	eleq1i 2243	. 2 ⊢ (∪ 𝒫 𝐴 ∈ V ↔ 𝐴 ∈ V)
4	1, 3	bitr2i 185	1 ⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)

Syntax hints:   ↔ wb 105   ∈ wcel 2148  Vcvv 2739  𝒫 cpw 3577  ∪ cuni 3811

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2741  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-uni 3812

This theorem is referenced by: (None)