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| 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 4638 | . . . 4 ⊢ {𝐵} = {𝐵, 𝐵} | |
| 2 | 1 | supeq1i 9488 | . . 3 ⊢ sup({𝐵}, 𝐴, 𝑅) = sup({𝐵, 𝐵}, 𝐴, 𝑅) | 
| 3 | suppr 9512 | . . . 4 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴) → sup({𝐵, 𝐵}, 𝐴, 𝑅) = if(𝐵𝑅𝐵, 𝐵, 𝐵)) | |
| 4 | 3 | 3anidm23 1422 | . . 3 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → sup({𝐵, 𝐵}, 𝐴, 𝑅) = if(𝐵𝑅𝐵, 𝐵, 𝐵)) | 
| 5 | 2, 4 | eqtrid 2788 | . 2 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → sup({𝐵}, 𝐴, 𝑅) = if(𝐵𝑅𝐵, 𝐵, 𝐵)) | 
| 6 | ifid 4565 | . 2 ⊢ if(𝐵𝑅𝐵, 𝐵, 𝐵) = 𝐵 | |
| 7 | 5, 6 | eqtrdi 2792 | 1 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → sup({𝐵}, 𝐴, 𝑅) = 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ifcif 4524 {csn 4625 {cpr 4627 class class class wbr 5142 Or wor 5590 supcsup 9481 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-po 5591 df-so 5592 df-iota 6513 df-riota 7389 df-sup 9483 | 
| This theorem is referenced by: supxrmnf 13360 ramz 17064 xpsdsval 24392 ovolctb 25526 nmoo0 30811 nmop0 32006 nmfn0 32007 esumnul 34050 esum0 34051 ovoliunnfl 37670 voliunnfl 37672 volsupnfl 37673 liminf10ex 45794 fourierdlem79 46205 sge0z 46395 sge00 46396 | 
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