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Theorem supsn 7220
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 3654 . . . 4  |-  { B }  =  { B ,  B }
21supeq1i 7200 . . 3  |-  sup ( { B } ,  A ,  R )  =  sup ( { B ,  B } ,  A ,  R )
3 suppr 7219 . . . 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 2327 . 2  |-  ( ( R  Or  A  /\  B  e.  A )  ->  sup ( { B } ,  A ,  R )  =  if ( B R B ,  B ,  B
) )
6 ifid 3597 . 2  |-  if ( B R B ,  B ,  B )  =  B
75, 6syl6eq 2331 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 1623    e. wcel 1684   ifcif 3565   {csn 3640   {cpr 3641   class class class wbr 4023    Or wor 4313   supcsup 7193
This theorem is referenced by:  supxrmnf  10636  ramz  13072  xpsdsval  17945  ovolctb  18849  nmoo0  21369  nmop0  22566  nmfn0  22567  esumnul  23427  esum0  23428
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-po 4314  df-so 4315  df-iota 5219  df-riota 6304  df-sup 7194
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