MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  somin2 Structured version   Visualization version   GIF version

Theorem somin2 5990
Description: Property of a minimum in a strict order. (Contributed by Stefan O'Rear, 17-Jan-2015.)
Assertion
Ref Expression
somin2 ((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) → if(𝐴𝑅𝐵, 𝐴, 𝐵)(𝑅 ∪ I )𝐵)

Proof of Theorem somin2
StepHypRef Expression
1 somincom 5989 . 2 ((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) → if(𝐴𝑅𝐵, 𝐴, 𝐵) = if(𝐵𝑅𝐴, 𝐵, 𝐴))
2 somin1 5988 . . 3 ((𝑅 Or 𝑋 ∧ (𝐵𝑋𝐴𝑋)) → if(𝐵𝑅𝐴, 𝐵, 𝐴)(𝑅 ∪ I )𝐵)
32ancom2s 648 . 2 ((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) → if(𝐵𝑅𝐴, 𝐵, 𝐴)(𝑅 ∪ I )𝐵)
41, 3eqbrtrd 5081 1 ((𝑅 Or 𝑋 ∧ (𝐴𝑋𝐵𝑋)) → if(𝐴𝑅𝐵, 𝐴, 𝐵)(𝑅 ∪ I )𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  wcel 2110  cun 3934  ifcif 4467   class class class wbr 5059   I cid 5454   Or wor 5468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pr 5322
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-br 5060  df-opab 5122  df-id 5455  df-po 5469  df-so 5470  df-xp 5556  df-rel 5557
This theorem is referenced by:  soltmin  5991
  Copyright terms: Public domain W3C validator