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Theorem supex2g 7008
Description: Existence of supremum. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
supex2g  |-  ( A  e.  C  ->  sup ( B ,  A ,  R )  e.  _V )

Proof of Theorem supex2g
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-sup 6959 . 2  |-  sup ( B ,  A ,  R )  =  U. { x  e.  A  |  ( A. y  e.  B  -.  x R y  /\  A. y  e.  A  (
y R x  ->  E. z  e.  B  y R z ) ) }
2 rabexg 4130 . . 3  |-  ( A  e.  C  ->  { x  e.  A  |  ( A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) ) }  e.  _V )
32uniexd 4423 . 2  |-  ( A  e.  C  ->  U. {
x  e.  A  | 
( A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R
z ) ) }  e.  _V )
41, 3eqeltrid 2257 1  |-  ( A  e.  C  ->  sup ( B ,  A ,  R )  e.  _V )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    e. wcel 2141   A.wral 2448   E.wrex 2449   {crab 2452   _Vcvv 2730   U.cuni 3794   class class class wbr 3987   supcsup 6957
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 4105  ax-un 4416
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-rab 2457  df-v 2732  df-in 3127  df-ss 3134  df-uni 3795  df-sup 6959
This theorem is referenced by:  infex2g  7009  pczpre  12244
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