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Theorem infsn 8565
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 4327 . . . 4 {𝐵} = {𝐵, 𝐵}
21infeq1i 8539 . . 3 inf({𝐵}, 𝐴, 𝑅) = inf({𝐵, 𝐵}, 𝐴, 𝑅)
3 infpr 8564 . . . 4 ((𝑅 Or 𝐴𝐵𝐴𝐵𝐴) → inf({𝐵, 𝐵}, 𝐴, 𝑅) = if(𝐵𝑅𝐵, 𝐵, 𝐵))
433anidm23 1530 . . 3 ((𝑅 Or 𝐴𝐵𝐴) → inf({𝐵, 𝐵}, 𝐴, 𝑅) = if(𝐵𝑅𝐵, 𝐵, 𝐵))
52, 4syl5eq 2816 . 2 ((𝑅 Or 𝐴𝐵𝐴) → inf({𝐵}, 𝐴, 𝑅) = if(𝐵𝑅𝐵, 𝐵, 𝐵))
6 ifid 4262 . 2 if(𝐵𝑅𝐵, 𝐵, 𝐵) = 𝐵
75, 6syl6eq 2820 1 ((𝑅 Or 𝐴𝐵𝐴) → inf({𝐵}, 𝐴, 𝑅) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1630  wcel 2144  ifcif 4223  {csn 4314  {cpr 4316   class class class wbr 4784   Or wor 5169  infcinf 8502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-po 5170  df-so 5171  df-cnv 5257  df-iota 5994  df-riota 6753  df-sup 8503  df-inf 8504
This theorem is referenced by:  infxrpnf  40184  limsup0  40438  limsuppnfdlem  40445  limsup10ex  40517
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