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Mirrors > Home > MPE Home > Th. List > supsn | Structured version Visualization version GIF version |
Description: The supremum of a singleton. (Contributed by NM, 2-Oct-2007.) |
Ref | Expression |
---|---|
supsn | ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → sup({𝐵}, 𝐴, 𝑅) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsn2 4644 | . . . 4 ⊢ {𝐵} = {𝐵, 𝐵} | |
2 | 1 | supeq1i 9485 | . . 3 ⊢ sup({𝐵}, 𝐴, 𝑅) = sup({𝐵, 𝐵}, 𝐴, 𝑅) |
3 | suppr 9509 | . . . 4 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴) → sup({𝐵, 𝐵}, 𝐴, 𝑅) = if(𝐵𝑅𝐵, 𝐵, 𝐵)) | |
4 | 3 | 3anidm23 1420 | . . 3 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → sup({𝐵, 𝐵}, 𝐴, 𝑅) = if(𝐵𝑅𝐵, 𝐵, 𝐵)) |
5 | 2, 4 | eqtrid 2787 | . 2 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → sup({𝐵}, 𝐴, 𝑅) = if(𝐵𝑅𝐵, 𝐵, 𝐵)) |
6 | ifid 4571 | . 2 ⊢ if(𝐵𝑅𝐵, 𝐵, 𝐵) = 𝐵 | |
7 | 5, 6 | eqtrdi 2791 | 1 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → sup({𝐵}, 𝐴, 𝑅) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ifcif 4531 {csn 4631 {cpr 4633 class class class wbr 5148 Or wor 5596 supcsup 9478 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-po 5597 df-so 5598 df-iota 6516 df-riota 7388 df-sup 9480 |
This theorem is referenced by: supxrmnf 13356 ramz 17059 xpsdsval 24407 ovolctb 25539 nmoo0 30820 nmop0 32015 nmfn0 32016 esumnul 34029 esum0 34030 ovoliunnfl 37649 voliunnfl 37651 volsupnfl 37652 liminf10ex 45730 fourierdlem79 46141 sge0z 46331 sge00 46332 |
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