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| Mirrors > Home > MPE Home > Th. List > infsn | Structured version Visualization version GIF version | ||
| Description: The infimum of a singleton. (Contributed by NM, 2-Oct-2007.) |
| Ref | Expression |
|---|---|
| infsn | ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → inf({𝐵}, 𝐴, 𝑅) = 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dfsn2 4639 | . . . 4 ⊢ {𝐵} = {𝐵, 𝐵} | |
| 2 | 1 | infeq1i 9518 | . . 3 ⊢ inf({𝐵}, 𝐴, 𝑅) = inf({𝐵, 𝐵}, 𝐴, 𝑅) |
| 3 | infpr 9543 | . . . 4 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴) → inf({𝐵, 𝐵}, 𝐴, 𝑅) = if(𝐵𝑅𝐵, 𝐵, 𝐵)) | |
| 4 | 3 | 3anidm23 1423 | . . 3 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → inf({𝐵, 𝐵}, 𝐴, 𝑅) = if(𝐵𝑅𝐵, 𝐵, 𝐵)) |
| 5 | 2, 4 | eqtrid 2789 | . 2 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → inf({𝐵}, 𝐴, 𝑅) = if(𝐵𝑅𝐵, 𝐵, 𝐵)) |
| 6 | ifid 4566 | . 2 ⊢ if(𝐵𝑅𝐵, 𝐵, 𝐵) = 𝐵 | |
| 7 | 5, 6 | eqtrdi 2793 | 1 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → inf({𝐵}, 𝐴, 𝑅) = 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ifcif 4525 {csn 4626 {cpr 4628 class class class wbr 5143 Or wor 5591 infcinf 9481 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-po 5592 df-so 5593 df-cnv 5693 df-iota 6514 df-riota 7388 df-sup 9482 df-inf 9483 |
| This theorem is referenced by: infxrpnf 45457 limsup0 45709 limsuppnfdlem 45716 limsup10ex 45788 |
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