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Theorem unimax 4226
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 3461 . . 3  |-  A  C_  A
2 sseq1 3463 . . . 4  |-  ( x  =  A  ->  (
x  C_  A  <->  A  C_  A
) )
32elrab3 3208 . . 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 3463 . . . . 5  |-  ( x  =  y  ->  (
x  C_  A  <->  y  C_  A ) )
65elrab 3207 . . . 4  |-  ( y  e.  { x  e.  B  |  x  C_  A }  <->  ( y  e.  B  /\  y  C_  A ) )
76simprbi 462 . . 3  |-  ( y  e.  { x  e.  B  |  x  C_  A }  ->  y  C_  A )
87rgen 2764 . 2  |-  A. y  e.  { x  e.  B  |  x  C_  A }
y  C_  A
9 ssunieq 4225 . . 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 2410 . 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 660 1  |-  ( A  e.  B  ->  U. {
x  e.  B  |  x  C_  A }  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   A.wral 2754   {crab 2758    C_ wss 3414   U.cuni 4191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2759  df-rab 2763  df-v 3061  df-in 3421  df-ss 3428  df-uni 4192
This theorem is referenced by:  lssuni  17906  chsupid  26744  shatomistici  27693  lssats  32030  lpssat  32031  lssatle  32033  lssat  32034
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