Theorem uniexb 4599

Description: The Axiom of Union and its converse. A class is a set iff its union is a set. (Contributed by NM, 11-Nov-2003.)

Assertion

Ref	Expression
uniexb	|- ( A e. _V <-> U. A e. _V )

Proof of Theorem uniexb

Step	Hyp	Ref	Expression
1		uniexg 4565	. 2 |- ( A e. _V -> U. A e. _V )
2		pwuni 4310	. . 3 |- A C_ ~P U. A
3		pwexg 4298	. . 3 |- ( U. A e. _V -> ~P U. A e. _V )
4		ssexg 4254	. . 3 |- ( ( A C_ ~P U. A /\ ~P U. A e. _V ) -> A e. _V )
5	2, 3, 4	sylancr 414	. 2 |- ( U. A e. _V -> A e. _V )
6	1, 5	impbii 126	1 |- ( A e. _V <-> U. A e. _V )

Syntax hints:   <-> wb 105   e. wcel 2205   _Vcvv 2815   C_ wss 3214   ~Pcpw 3674   U.cuni 3919

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rex 2528  df-v 2817  df-in 3220  df-ss 3227  df-pw 3676  df-uni 3920

This theorem is referenced by:  pwexb  4600  elpwpwel  4601  tfrlemibex  6573  tfr1onlembex  6589  tfrcllembex  6602  ixpexgg  6970  ptex  13561  tgss2  15070  txbasex  15248