Theorem maxcom 11969

Description: The maximum of two reals is commutative. Lemma 3.9 of [Geuvers], p. 10. (Contributed by Jim Kingdon, 21-Dec-2021.)

Assertion

Ref	Expression
maxcom	|- sup ( { A , B } , RR , < ) = sup ( { B , A } , RR , < )

Proof of Theorem maxcom

Step	Hyp	Ref	Expression
1		prcom 3787	. 2 |- { A , B } = { B , A }
2	1	supeq1i 7328	1 |- sup ( { A , B } , RR , < ) = sup ( { B , A } , RR , < )

This proof depends on syntax axioms:   = wceq 1402   {cpr 3710   supcsup 7322   RRcr 8178   < clt 8360

This proof depends on 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 proof 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 3716  df-uni 3936  df-sup 7324

This theorem is used by:  maxle2  11978  maxclpr  11988  2zsupmax  11992  xrmaxiflemcom  12015  repiecege0  17076