Theorem maxcom 11680

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 3722	. 2 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴}
2	1	supeq1i 7123	1 ⊢ sup({𝐴, 𝐵}, ℝ, < ) = sup({𝐵, 𝐴}, ℝ, < )

Syntax hints:   = wceq 1375  {cpr 3647  supcsup 7117  ℝcr 7966   < clt 8149

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-ext 2191

This theorem depends on definitions:  df-bi 117  df-tru 1378  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ral 2493  df-rex 2494  df-rab 2497  df-v 2781  df-un 3181  df-pr 3653  df-uni 3868  df-sup 7119

This theorem is referenced by:  maxle2  11689  maxclpr  11699  2zsupmax  11703  xrmaxiflemcom  11726