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Theorem mincom 11269
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 3683 . 2 {𝐴, 𝐵} = {𝐵, 𝐴}
21infeq1i 7042 1 inf({𝐴, 𝐵}, ℝ, < ) = inf({𝐵, 𝐴}, ℝ, < )
Colors of variables: wff set class
Syntax hints:   = wceq 1364  {cpr 3608  infcinf 7012  cr 7840   < clt 8022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-un 3148  df-pr 3614  df-uni 3825  df-sup 7013  df-inf 7014
This theorem is referenced by:  mingeb  11282  2zinfmin  11283
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