Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  supex2g Unicode version

Theorem supex2g 25743
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 7284 . 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 4245 . . 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 )
3 uniexg 4599 . . 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 )
42, 3syl 15 . 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 )
51, 4syl5eqel 2442 1  |-  ( A  e.  C  ->  sup ( B ,  A ,  R )  e.  _V )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    e. wcel 1710   A.wral 2619   E.wrex 2620   {crab 2623   _Vcvv 2864   U.cuni 3908   class class class wbr 4104   supcsup 7283
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-rex 2625  df-rab 2628  df-v 2866  df-in 3235  df-ss 3242  df-uni 3909  df-sup 7284
  Copyright terms: Public domain W3C validator