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Theorem maxcom 10290
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
StepHypRef Expression
1 prcom 3486 . 2 {𝐴, 𝐵} = {𝐵, 𝐴}
21supeq1i 6495 1 sup({𝐴, 𝐵}, ℝ, < ) = sup({𝐵, 𝐴}, ℝ, < )
Colors of variables: wff set class
Syntax hints:   = wceq 1285  {cpr 3417  supcsup 6489  cr 7094   < clt 7267
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2612  df-un 2986  df-pr 3423  df-uni 3622  df-sup 6491
This theorem is referenced by:  maxle2  10299  maxclpr  10309
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