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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 3635 | . 2 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴} | |
2 | 1 | infeq1i 6957 | 1 ⊢ inf({𝐴, 𝐵}, ℝ, < ) = inf({𝐵, 𝐴}, ℝ, < ) |
Colors of variables: wff set class |
Syntax hints: = wceq 1335 {cpr 3561 infcinf 6927 ℝcr 7731 < clt 7912 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-un 3106 df-pr 3567 df-uni 3773 df-sup 6928 df-inf 6929 |
This theorem is referenced by: (None) |
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