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Theorem unimax 4274
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 3516 . . 3  |-  A  C_  A
2 sseq1 3518 . . . 4  |-  ( x  =  A  ->  (
x  C_  A  <->  A  C_  A
) )
32elrab3 3255 . . 3  |-  ( A  e.  B  ->  ( A  e.  { x  e.  B  |  x  C_  A }  <->  A  C_  A
) )
41, 3mpbiri 233 . 2  |-  ( A  e.  B  ->  A  e.  { x  e.  B  |  x  C_  A }
)
5 sseq1 3518 . . . . 5  |-  ( x  =  y  ->  (
x  C_  A  <->  y  C_  A ) )
65elrab 3254 . . . 4  |-  ( y  e.  { x  e.  B  |  x  C_  A }  <->  ( y  e.  B  /\  y  C_  A ) )
76simprbi 464 . . 3  |-  ( y  e.  { x  e.  B  |  x  C_  A }  ->  y  C_  A )
87rgen 2817 . 2  |-  A. y  e.  { x  e.  B  |  x  C_  A }
y  C_  A
9 ssunieq 4273 . . 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 2468 . 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 662 1  |-  ( A  e.  B  ->  U. {
x  e.  B  |  x  C_  A }  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   A.wral 2807   {crab 2811    C_ wss 3469   U.cuni 4238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ral 2812  df-rab 2816  df-v 3108  df-in 3476  df-ss 3483  df-uni 4239
This theorem is referenced by:  lssuni  17362  chsupid  25992  shatomistici  26942  lssats  33684  lpssat  33685  lssatle  33687  lssat  33688
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