Theorem uniexb 4564

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 4530	. 2 |- ( A e. _V -> U. A e. _V )
2		pwuni 4276	. . 3 |- A C_ ~P U. A
3		pwexg 4264	. . 3 |- ( U. A e. _V -> ~P U. A e. _V )
4		ssexg 4223	. . 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 2200   _Vcvv 2799   C_ wss 3197   ~Pcpw 3649   U.cuni 3888

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-un 4524

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2801  df-in 3203  df-ss 3210  df-pw 3651  df-uni 3889

This theorem is referenced by:  pwexb  4565  elpwpwel  4566  tfrlemibex  6475  tfr1onlembex  6491  tfrcllembex  6504  ixpexgg  6869  ptex  13297  tgss2  14753  txbasex  14931