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Theorem unimax 3877
Description: Any member of a class is the largest of those members that it includes. (Contributed by NM, 13-Aug-2002.)
Assertion
Ref Expression
unimax  |-  ( A  e.  B  ->  U. {
x  e.  B  |  x  C_  A }  =  A )
Distinct variable groups:    x, A    x, B

Proof of Theorem unimax
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ssid 3210 . . 3  |-  A  C_  A
2 sseq1 3212 . . . 4  |-  ( x  =  A  ->  (
x  C_  A  <->  A  C_  A
) )
32elrab3 2937 . . 3  |-  ( A  e.  B  ->  ( A  e.  { x  e.  B  |  x  C_  A }  <->  A  C_  A
) )
41, 3mpbiri 224 . 2  |-  ( A  e.  B  ->  A  e.  { x  e.  B  |  x  C_  A }
)
5 sseq1 3212 . . . . 5  |-  ( x  =  y  ->  (
x  C_  A  <->  y  C_  A ) )
65elrab 2936 . . . 4  |-  ( y  e.  { x  e.  B  |  x  C_  A }  <->  ( y  e.  B  /\  y  C_  A ) )
76simprbi 450 . . 3  |-  ( y  e.  { x  e.  B  |  x  C_  A }  ->  y  C_  A )
87rgen 2621 . 2  |-  A. y  e.  { x  e.  B  |  x  C_  A }
y  C_  A
9 ssunieq 3876 . . 3  |-  ( ( A  e.  { x  e.  B  |  x  C_  A }  /\  A. y  e.  { x  e.  B  |  x  C_  A } y  C_  A )  ->  A  =  U. { x  e.  B  |  x  C_  A } )
109eqcomd 2301 . 2  |-  ( ( A  e.  { x  e.  B  |  x  C_  A }  /\  A. y  e.  { x  e.  B  |  x  C_  A } y  C_  A )  ->  U. {
x  e.  B  |  x  C_  A }  =  A )
114, 8, 10sylancl 643 1  |-  ( A  e.  B  ->  U. {
x  e.  B  |  x  C_  A }  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560    C_ wss 3165   U.cuni 3843
This theorem is referenced by:  lssuni  15713  chsupid  22007  shatomistici  22957  lssats  29824  lpssat  29825  lssatle  29827  lssat  29828
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rab 2565  df-v 2803  df-in 3172  df-ss 3179  df-uni 3844
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