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Theorem supsn 8322
Description: The supremum of a singleton. (Contributed by NM, 2-Oct-2007.)
Assertion
Ref Expression
supsn ((𝑅 Or 𝐴𝐵𝐴) → sup({𝐵}, 𝐴, 𝑅) = 𝐵)

Proof of Theorem supsn
StepHypRef Expression
1 dfsn2 4161 . . . 4 {𝐵} = {𝐵, 𝐵}
21supeq1i 8297 . . 3 sup({𝐵}, 𝐴, 𝑅) = sup({𝐵, 𝐵}, 𝐴, 𝑅)
3 suppr 8321 . . . 4 ((𝑅 Or 𝐴𝐵𝐴𝐵𝐴) → sup({𝐵, 𝐵}, 𝐴, 𝑅) = if(𝐵𝑅𝐵, 𝐵, 𝐵))
433anidm23 1382 . . 3 ((𝑅 Or 𝐴𝐵𝐴) → sup({𝐵, 𝐵}, 𝐴, 𝑅) = if(𝐵𝑅𝐵, 𝐵, 𝐵))
52, 4syl5eq 2667 . 2 ((𝑅 Or 𝐴𝐵𝐴) → sup({𝐵}, 𝐴, 𝑅) = if(𝐵𝑅𝐵, 𝐵, 𝐵))
6 ifid 4097 . 2 if(𝐵𝑅𝐵, 𝐵, 𝐵) = 𝐵
75, 6syl6eq 2671 1 ((𝑅 Or 𝐴𝐵𝐴) → sup({𝐵}, 𝐴, 𝑅) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  ifcif 4058  {csn 4148  {cpr 4150   class class class wbr 4613   Or wor 4994  supcsup 8290
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-po 4995  df-so 4996  df-iota 5810  df-riota 6565  df-sup 8292
This theorem is referenced by:  supxrmnf  12090  ramz  15653  xpsdsval  22096  ovolctb  23165  nmoo0  27492  nmop0  28691  nmfn0  28692  esumnul  29888  esum0  29889  ovoliunnfl  33080  voliunnfl  33082  volsupnfl  33083  fourierdlem79  39706  sge0z  39896  sge00  39897
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