Theorem uniexb 4619

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 4585	. 2 |- ( A e. _V -> U. A e. _V )
2		pwuni 4329	. . 3 |- A C_ ~P U. A
3		pwexg 4317	. . 3 |- ( U. A e. _V -> ~P U. A e. _V )
4		ssexg 4272	. . 3 |- ( ( A C_ ~P U. A /\ ~P U. A e. _V ) -> A e. _V )
5	2, 3, 4	sylancr 418	. 2 |- ( U. A e. _V -> A e. _V )
6	1, 5	impbii 126	1 |- ( A e. _V <-> U. A e. _V )

This proof depends on syntax axioms:   <-> wb 105   e. wcel 2209   _Vcvv 2821   C_ wss 3220   ~Pcpw 3688   U.cuni 3935

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-un 4578

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-in 3226  df-ss 3233  df-pw 3690  df-uni 3936

This theorem is used by:  pwexb  4620  elpwpwel  4621  tfrlemibex  6600  tfr1onlembex  6616  tfrcllembex  6629  ixpexgg  7004  ptex  13618  tgss2  15180  txbasex  15358