ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  uniexb Unicode version

Theorem uniexb 4458
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
StepHypRef Expression
1 uniexg 4424 . 2  |-  ( A  e.  _V  ->  U. A  e.  _V )
2 pwuni 4178 . . 3  |-  A  C_  ~P U. A
3 pwexg 4166 . . 3  |-  ( U. A  e.  _V  ->  ~P
U. A  e.  _V )
4 ssexg 4128 . . 3  |-  ( ( A  C_  ~P U. A  /\  ~P U. A  e. 
_V )  ->  A  e.  _V )
52, 3, 4sylancr 412 . 2  |-  ( U. A  e.  _V  ->  A  e.  _V )
61, 5impbii 125 1  |-  ( A  e.  _V  <->  U. A  e. 
_V )
Colors of variables: wff set class
Syntax hints:    <-> wb 104    e. wcel 2141   _Vcvv 2730    C_ wss 3121   ~Pcpw 3566   U.cuni 3796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-un 4418
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-in 3127  df-ss 3134  df-pw 3568  df-uni 3797
This theorem is referenced by:  pwexb  4459  elpwpwel  4460  tfrlemibex  6308  tfr1onlembex  6324  tfrcllembex  6337  ixpexgg  6700  tgss2  12873  txbasex  13051
  Copyright terms: Public domain W3C validator