Theorem mincom 11869

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 3751	. 2 |- { A , B } = { B , A }
2	1	infeq1i 7272	1 |- inf ( { A , B } , RR , < ) = inf ( { B , A } , RR , < )

Syntax hints:   = wceq 1398   {cpr 3674  infcinf 7242   RRcr 8091   < clt 8273

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-un 3205  df-pr 3680  df-uni 3899  df-sup 7243  df-inf 7244

This theorem is referenced by:  mingeb  11882  2zinfmin  11883  hovergt0  15461  repiecege0  16759