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Theorem unimax 3765
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 3112 . . 3  |-  A  C_  A
2 sseq1 3115 . . . 4  |-  ( x  =  A  ->  (
x  C_  A  <->  A  C_  A
) )
32elrab3 2836 . . 3  |-  ( A  e.  B  ->  ( A  e.  { x  e.  B  |  x  C_  A }  <->  A  C_  A
) )
41, 3mpbiri 167 . 2  |-  ( A  e.  B  ->  A  e.  { x  e.  B  |  x  C_  A }
)
5 sseq1 3115 . . . . 5  |-  ( x  =  y  ->  (
x  C_  A  <->  y  C_  A ) )
65elrab 2835 . . . 4  |-  ( y  e.  { x  e.  B  |  x  C_  A }  <->  ( y  e.  B  /\  y  C_  A ) )
76simprbi 273 . . 3  |-  ( y  e.  { x  e.  B  |  x  C_  A }  ->  y  C_  A )
87rgen 2483 . 2  |-  A. y  e.  { x  e.  B  |  x  C_  A }
y  C_  A
9 ssunieq 3764 . . 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 2143 . 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 409 1  |-  ( A  e.  B  ->  U. {
x  e.  B  |  x  C_  A }  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480   A.wral 2414   {crab 2418    C_ wss 3066   U.cuni 3731
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rab 2423  df-v 2683  df-in 3072  df-ss 3079  df-uni 3732
This theorem is referenced by:  onuniss2  4423
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