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Theorem supsn 7236
Description: The supremum of a singleton. (Contributed by NM, 2-Oct-2007.)
Assertion
Ref Expression
supsn  |-  ( ( R  Or  A  /\  B  e.  A )  ->  sup ( { B } ,  A ,  R )  =  B )

Proof of Theorem supsn
StepHypRef Expression
1 dfsn2 3667 . . . 4  |-  { B }  =  { B ,  B }
21supeq1i 7216 . . 3  |-  sup ( { B } ,  A ,  R )  =  sup ( { B ,  B } ,  A ,  R )
3 suppr 7235 . . . 4  |-  ( ( R  Or  A  /\  B  e.  A  /\  B  e.  A )  ->  sup ( { B ,  B } ,  A ,  R )  =  if ( B R B ,  B ,  B
) )
433anidm23 1241 . . 3  |-  ( ( R  Or  A  /\  B  e.  A )  ->  sup ( { B ,  B } ,  A ,  R )  =  if ( B R B ,  B ,  B
) )
52, 4syl5eq 2340 . 2  |-  ( ( R  Or  A  /\  B  e.  A )  ->  sup ( { B } ,  A ,  R )  =  if ( B R B ,  B ,  B
) )
6 ifid 3610 . 2  |-  if ( B R B ,  B ,  B )  =  B
75, 6syl6eq 2344 1  |-  ( ( R  Or  A  /\  B  e.  A )  ->  sup ( { B } ,  A ,  R )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   ifcif 3578   {csn 3653   {cpr 3654   class class class wbr 4039    Or wor 4329   supcsup 7209
This theorem is referenced by:  supxrmnf  10652  ramz  13088  xpsdsval  17961  ovolctb  18865  nmoo0  21385  nmop0  22582  nmfn0  22583  esumnul  23442  esum0  23443  ovoliunnfl  25001
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-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-po 4330  df-so 4331  df-iota 5235  df-riota 6320  df-sup 7210
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