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Theorem infsn 8293
 Description: The infimum of a singleton. (Contributed by NM, 2-Oct-2007.)
Assertion
Ref Expression
infsn ((𝑅 Or 𝐴𝐵𝐴) → inf({𝐵}, 𝐴, 𝑅) = 𝐵)

Proof of Theorem infsn
StepHypRef Expression
1 dfsn2 4138 . . . 4 {𝐵} = {𝐵, 𝐵}
21infeq1i 8267 . . 3 inf({𝐵}, 𝐴, 𝑅) = inf({𝐵, 𝐵}, 𝐴, 𝑅)
3 infpr 8292 . . . 4 ((𝑅 Or 𝐴𝐵𝐴𝐵𝐴) → inf({𝐵, 𝐵}, 𝐴, 𝑅) = if(𝐵𝑅𝐵, 𝐵, 𝐵))
433anidm23 1377 . . 3 ((𝑅 Or 𝐴𝐵𝐴) → inf({𝐵, 𝐵}, 𝐴, 𝑅) = if(𝐵𝑅𝐵, 𝐵, 𝐵))
52, 4syl5eq 2656 . 2 ((𝑅 Or 𝐴𝐵𝐴) → inf({𝐵}, 𝐴, 𝑅) = if(𝐵𝑅𝐵, 𝐵, 𝐵))
6 ifid 4075 . 2 if(𝐵𝑅𝐵, 𝐵, 𝐵) = 𝐵
75, 6syl6eq 2660 1 ((𝑅 Or 𝐴𝐵𝐴) → inf({𝐵}, 𝐴, 𝑅) = 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ifcif 4036  {csn 4125  {cpr 4127   class class class wbr 4583   Or wor 4958  infcinf 8230 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-po 4959  df-so 4960  df-cnv 5046  df-iota 5768  df-riota 6511  df-sup 8231  df-inf 8232 This theorem is referenced by: (None)
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