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Theorem supsn 8545
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 4334 . . . 4 {𝐵} = {𝐵, 𝐵}
21supeq1i 8520 . . 3 sup({𝐵}, 𝐴, 𝑅) = sup({𝐵, 𝐵}, 𝐴, 𝑅)
3 suppr 8544 . . . 4 ((𝑅 Or 𝐴𝐵𝐴𝐵𝐴) → sup({𝐵, 𝐵}, 𝐴, 𝑅) = if(𝐵𝑅𝐵, 𝐵, 𝐵))
433anidm23 1532 . . 3 ((𝑅 Or 𝐴𝐵𝐴) → sup({𝐵, 𝐵}, 𝐴, 𝑅) = if(𝐵𝑅𝐵, 𝐵, 𝐵))
52, 4syl5eq 2806 . 2 ((𝑅 Or 𝐴𝐵𝐴) → sup({𝐵}, 𝐴, 𝑅) = if(𝐵𝑅𝐵, 𝐵, 𝐵))
6 ifid 4269 . 2 if(𝐵𝑅𝐵, 𝐵, 𝐵) = 𝐵
75, 6syl6eq 2810 1 ((𝑅 Or 𝐴𝐵𝐴) → sup({𝐵}, 𝐴, 𝑅) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  ifcif 4230  {csn 4321  {cpr 4323   class class class wbr 4804   Or wor 5186  supcsup 8513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-po 5187  df-so 5188  df-iota 6012  df-riota 6775  df-sup 8515
This theorem is referenced by:  supxrmnf  12360  ramz  15951  xpsdsval  22407  ovolctb  23478  nmoo0  27976  nmop0  29175  nmfn0  29176  esumnul  30440  esum0  30441  ovoliunnfl  33782  voliunnfl  33784  volsupnfl  33785  liminf10ex  40527  fourierdlem79  40923  sge0z  41113  sge00  41114
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