Theorem maxcom 11765

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 3747	. 2 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴}
2	1	supeq1i 7187	1 ⊢ sup({𝐴, 𝐵}, ℝ, < ) = sup({𝐵, 𝐴}, ℝ, < )

Syntax hints:   = wceq 1397  {cpr 3670  supcsup 7181  ℝcr 8031   < clt 8214

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-un 3204  df-pr 3676  df-uni 3894  df-sup 7183

This theorem is referenced by:  maxle2  11774  maxclpr  11784  2zsupmax  11788  xrmaxiflemcom  11811