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Mirrors > Home > ILE Home > Th. List > mincom | GIF version |
Description: The minimum of two reals is commutative. (Contributed by Jim Kingdon, 8-Feb-2021.) |
Ref | Expression |
---|---|
mincom | ⊢ inf({𝐴, 𝐵}, ℝ, < ) = inf({𝐵, 𝐴}, ℝ, < ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prcom 3563 | . 2 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴} | |
2 | 1 | infeq1i 6850 | 1 ⊢ inf({𝐴, 𝐵}, ℝ, < ) = inf({𝐵, 𝐴}, ℝ, < ) |
Colors of variables: wff set class |
Syntax hints: = wceq 1312 {cpr 3492 infcinf 6820 ℝcr 7540 < clt 7718 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 |
This theorem depends on definitions: df-bi 116 df-tru 1315 df-nf 1418 df-sb 1717 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ral 2393 df-rex 2394 df-rab 2397 df-v 2657 df-un 3039 df-pr 3498 df-uni 3701 df-sup 6821 df-inf 6822 |
This theorem is referenced by: (None) |
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