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Theorem unimax 3614
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 2964 . . 3  |-  A  C_  A
2 sseq1 2966 . . . 4  |-  ( x  =  A  ->  (
x  C_  A  <->  A  C_  A
) )
32elrab3 2699 . . 3  |-  ( A  e.  B  ->  ( A  e.  { x  e.  B  |  x  C_  A }  <->  A  C_  A
) )
41, 3mpbiri 157 . 2  |-  ( A  e.  B  ->  A  e.  { x  e.  B  |  x  C_  A }
)
5 sseq1 2966 . . . . 5  |-  ( x  =  y  ->  (
x  C_  A  <->  y  C_  A ) )
65elrab 2698 . . . 4  |-  ( y  e.  { x  e.  B  |  x  C_  A }  <->  ( y  e.  B  /\  y  C_  A ) )
76simprbi 260 . . 3  |-  ( y  e.  { x  e.  B  |  x  C_  A }  ->  y  C_  A )
87rgen 2374 . 2  |-  A. y  e.  { x  e.  B  |  x  C_  A }
y  C_  A
9 ssunieq 3613 . . 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 2045 . 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 392 1  |-  ( A  e.  B  ->  U. {
x  e.  B  |  x  C_  A }  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    = wceq 1243    e. wcel 1393   A.wral 2306   {crab 2310    C_ wss 2917   U.cuni 3580
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rab 2315  df-v 2559  df-in 2924  df-ss 2931  df-uni 3581
This theorem is referenced by:  onuniss2  4238
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