Theorem maxcom 11080

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 3631	. 2 |- { A , B } = { B , A }
2	1	supeq1i 6920	1 |- sup ( { A , B } , RR , < ) = sup ( { B , A } , RR , < )

Syntax hints:   = wceq 1332   {cpr 3557   supcsup 6914   RRcr 7710   < clt 7891

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-rex 2438  df-rab 2441  df-v 2711  df-un 3102  df-pr 3563  df-uni 3769  df-sup 6916

This theorem is referenced by:  maxle2  11089  maxclpr  11099  2zsupmax  11102  xrmaxiflemcom  11123