Theorem mincom 10623

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

Step	Hyp	Ref	Expression
1		prcom 3513	. 2 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴}
2	1	infeq1i 6687	1 ⊢ inf({𝐴, 𝐵}, ℝ, < ) = inf({𝐵, 𝐴}, ℝ, < )

Syntax hints:   = wceq 1289  {cpr 3442  infcinf 6657  ℝcr 7328   < clt 7501

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-un 3001  df-pr 3448  df-uni 3649  df-sup 6658  df-inf 6659

This theorem is referenced by: (None)