Theorem mincom 11973

Description: The minimum of two reals is commutative. (Contributed by Jim Kingdon, 8-Feb-2021.)

Assertion

Ref	Expression
mincom	|- inf ( { A , B } , RR , < ) = inf ( { B , A } , RR , < )

Proof of Theorem mincom

Step	Hyp	Ref	Expression
1		prcom 3783	. 2 |- { A , B } = { B , A }
2	1	infeq1i 7343	1 |- inf ( { A , B } , RR , < ) = inf ( { B , A } , RR , < )

Syntax hints:   = wceq 1402   {cpr 3706  infcinf 7313   RRcr 8168   < clt 8350

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-un 3224  df-pr 3712  df-uni 3931  df-sup 7314  df-inf 7315

This theorem is referenced by:  mingeb  11986  2zinfmin  11987  hovergt0  15674  repiecege0  16981