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Theorem supexd 7422
Description: A supremum is a set. (Contributed by NM, 22-May-1999.) (Revised by Mario Carneiro, 24-Dec-2016.)
Hypothesis
Ref Expression
supmo.1  |-  ( ph  ->  R  Or  A )
Assertion
Ref Expression
supexd  |-  ( ph  ->  sup ( B ,  A ,  R )  e.  _V )

Proof of Theorem supexd
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-sup 7412 . 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 supmo.1 . . . 4  |-  ( ph  ->  R  Or  A )
32supmo 7421 . . 3  |-  ( ph  ->  E* x  e.  A
( A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R
z ) ) )
4 rmorabex 4391 . . 3  |-  ( E* x  e.  A ( A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) )  ->  { 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 )
5 uniexg 4673 . . 3  |-  ( { 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  ->  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 )
63, 4, 53syl 19 . 2  |-  ( ph  ->  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 )
71, 6syl5eqel 2496 1  |-  ( ph  ->  sup ( B ,  A ,  R )  e.  _V )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    e. wcel 1721   A.wral 2674   E.wrex 2675   E*wrmo 2677   {crab 2678   _Vcvv 2924   U.cuni 3983   class class class wbr 4180    Or wor 4470   supcsup 7411
This theorem is referenced by:  supex  7432
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rmo 2682  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-po 4471  df-so 4472  df-sup 7412
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