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Mirrors > Home > MPE Home > Th. List > somin2 | Structured version Visualization version GIF version |
Description: Property of a minimum in a strict order. (Contributed by Stefan O'Rear, 17-Jan-2015.) |
Ref | Expression |
---|---|
somin2 | ⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → if(𝐴𝑅𝐵, 𝐴, 𝐵)(𝑅 ∪ I )𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | somincom 6135 | . 2 ⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → if(𝐴𝑅𝐵, 𝐴, 𝐵) = if(𝐵𝑅𝐴, 𝐵, 𝐴)) | |
2 | somin1 6134 | . . 3 ⊢ ((𝑅 Or 𝑋 ∧ (𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → if(𝐵𝑅𝐴, 𝐵, 𝐴)(𝑅 ∪ I )𝐵) | |
3 | 2 | ancom2s 648 | . 2 ⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → if(𝐵𝑅𝐴, 𝐵, 𝐴)(𝑅 ∪ I )𝐵) |
4 | 1, 3 | eqbrtrd 5170 | 1 ⊢ ((𝑅 Or 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → if(𝐴𝑅𝐵, 𝐴, 𝐵)(𝑅 ∪ I )𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2106 ∪ cun 3946 ifcif 4528 class class class wbr 5148 I cid 5573 Or wor 5587 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 |
This theorem is referenced by: soltmin 6137 |
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