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Mirrors > Home > ILE Home > Th. List > maxcom | GIF version |
Description: The maximum of two reals is commutative. Lemma 3.9 of [Geuvers], p. 10. (Contributed by Jim Kingdon, 21-Dec-2021.) |
Ref | Expression |
---|---|
maxcom | ⊢ sup({𝐴, 𝐵}, ℝ, < ) = sup({𝐵, 𝐴}, ℝ, < ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prcom 3659 | . 2 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴} | |
2 | 1 | supeq1i 6965 | 1 ⊢ sup({𝐴, 𝐵}, ℝ, < ) = sup({𝐵, 𝐴}, ℝ, < ) |
Colors of variables: wff set class |
Syntax hints: = wceq 1348 {cpr 3584 supcsup 6959 ℝcr 7773 < clt 7954 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-un 3125 df-pr 3590 df-uni 3797 df-sup 6961 |
This theorem is referenced by: maxle2 11176 maxclpr 11186 2zsupmax 11189 xrmaxiflemcom 11212 |
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