Theorem maxcom 11754

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 3745	. 2 |- { A , B } = { B , A }
2	1	supeq1i 7178	1 |- sup ( { A , B } , RR , < ) = sup ( { B , A } , RR , < )

Syntax hints:   = wceq 1395   {cpr 3668   supcsup 7172   RRcr 8021   < clt 8204

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2802  df-un 3202  df-pr 3674  df-uni 3892  df-sup 7174

This theorem is referenced by:  maxle2  11763  maxclpr  11773  2zsupmax  11777  xrmaxiflemcom  11800