MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  infsn Structured version   Visualization version   GIF version

Theorem infsn 9545
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 4639 . . . 4 {𝐵} = {𝐵, 𝐵}
21infeq1i 9518 . . 3 inf({𝐵}, 𝐴, 𝑅) = inf({𝐵, 𝐵}, 𝐴, 𝑅)
3 infpr 9543 . . . 4 ((𝑅 Or 𝐴𝐵𝐴𝐵𝐴) → inf({𝐵, 𝐵}, 𝐴, 𝑅) = if(𝐵𝑅𝐵, 𝐵, 𝐵))
433anidm23 1423 . . 3 ((𝑅 Or 𝐴𝐵𝐴) → inf({𝐵, 𝐵}, 𝐴, 𝑅) = if(𝐵𝑅𝐵, 𝐵, 𝐵))
52, 4eqtrid 2789 . 2 ((𝑅 Or 𝐴𝐵𝐴) → inf({𝐵}, 𝐴, 𝑅) = if(𝐵𝑅𝐵, 𝐵, 𝐵))
6 ifid 4566 . 2 if(𝐵𝑅𝐵, 𝐵, 𝐵) = 𝐵
75, 6eqtrdi 2793 1 ((𝑅 Or 𝐴𝐵𝐴) → inf({𝐵}, 𝐴, 𝑅) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  ifcif 4525  {csn 4626  {cpr 4628   class class class wbr 5143   Or wor 5591  infcinf 9481
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-po 5592  df-so 5593  df-cnv 5693  df-iota 6514  df-riota 7388  df-sup 9482  df-inf 9483
This theorem is referenced by:  infxrpnf  45457  limsup0  45709  limsuppnfdlem  45716  limsup10ex  45788
  Copyright terms: Public domain W3C validator