Theorem uniexr 6972

Description: Converse of the Axiom of Union. Note that it does not require ax-un 6949. (Contributed by NM, 11-Nov-2003.)

Assertion

Ref	Expression
uniexr	|- ( U. A e. V -> A e. _V )

Proof of Theorem uniexr

Step	Hyp	Ref	Expression
1		pwuni 4474	. 2 |- A C_ ~P U. A
2		pwexg 4850	. 2 |- ( U. A e. V -> ~P U. A e. _V )
3		ssexg 4804	. 2 |- ( ( A C_ ~P U. A /\ ~P U. A e. _V ) -> A e. _V )
4	1, 2, 3	sylancr 695	1 |- ( U. A e. V -> A e. _V )

Syntax hints:   -> wi 4   e. wcel 1990   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   U.cuni 4436

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  ax-sep 4781  ax-pow 4843

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  df-uni 4437

This theorem is referenced by:  uniexb  6973  ssonprc  6992  ac5num  8859  bj-restv  33048  bj-mooreset  33056