Theorem mincom 11914

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 3767	. 2 |- { A , B } = { B , A }
2	1	infeq1i 7304	1 |- inf ( { A , B } , RR , < ) = inf ( { B , A } , RR , < )

Syntax hints:   = wceq 1398   {cpr 3690  infcinf 7274   RRcr 8126   < clt 8308

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 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-un 3215  df-pr 3696  df-uni 3915  df-sup 7275  df-inf 7276

This theorem is referenced by:  mingeb  11927  2zinfmin  11928  hovergt0  15515  repiecege0  16811