ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  mincom GIF version

Theorem mincom 11192
Description: The minimum of two reals is commutative. (Contributed by Jim Kingdon, 8-Feb-2021.)
Assertion
Ref Expression
mincom inf({𝐴, 𝐵}, ℝ, < ) = inf({𝐵, 𝐴}, ℝ, < )

Proof of Theorem mincom
StepHypRef Expression
1 prcom 3659 . 2 {𝐴, 𝐵} = {𝐵, 𝐴}
21infeq1i 6990 1 inf({𝐴, 𝐵}, ℝ, < ) = inf({𝐵, 𝐴}, ℝ, < )
Colors of variables: wff set class
Syntax hints:   = wceq 1348  {cpr 3584  infcinf 6960  cr 7773   < clt 7954
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-un 3125  df-pr 3590  df-uni 3797  df-sup 6961  df-inf 6962
This theorem is referenced by:  mingeb  11205  2zinfmin  11206
  Copyright terms: Public domain W3C validator