Theorem maxcom 11916

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}, ℝ, < ) = sup({𝐵, 𝐴}, ℝ, < )

Proof of Theorem maxcom

Step	Hyp	Ref	Expression
1		prcom 3772	. 2 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴}
2	1	supeq1i 7292	1 ⊢ sup({𝐴, 𝐵}, ℝ, < ) = sup({𝐵, 𝐴}, ℝ, < )

Syntax hints:   = wceq 1398  {cpr 3695  supcsup 7286  ℝcr 8142   < clt 8324

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 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-un 3218  df-pr 3701  df-uni 3920  df-sup 7288

This theorem is referenced by:  maxle2  11925  maxclpr  11935  2zsupmax  11939  xrmaxiflemcom  11962  repiecege0  16950