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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 4330 | . . . 4 ⊢ {𝐵} = {𝐵, 𝐵} | |
2 | 1 | infeq1i 8544 | . . 3 ⊢ inf({𝐵}, 𝐴, 𝑅) = inf({𝐵, 𝐵}, 𝐴, 𝑅) |
3 | infpr 8569 | . . . 4 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴) → inf({𝐵, 𝐵}, 𝐴, 𝑅) = if(𝐵𝑅𝐵, 𝐵, 𝐵)) | |
4 | 3 | 3anidm23 1531 | . . 3 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → inf({𝐵, 𝐵}, 𝐴, 𝑅) = if(𝐵𝑅𝐵, 𝐵, 𝐵)) |
5 | 2, 4 | syl5eq 2817 | . 2 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → inf({𝐵}, 𝐴, 𝑅) = if(𝐵𝑅𝐵, 𝐵, 𝐵)) |
6 | ifid 4265 | . 2 ⊢ if(𝐵𝑅𝐵, 𝐵, 𝐵) = 𝐵 | |
7 | 5, 6 | syl6eq 2821 | 1 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → inf({𝐵}, 𝐴, 𝑅) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ifcif 4226 {csn 4317 {cpr 4319 class class class wbr 4787 Or wor 5170 infcinf 8507 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pr 5035 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4227 df-sn 4318 df-pr 4320 df-op 4324 df-uni 4576 df-br 4788 df-opab 4848 df-po 5171 df-so 5172 df-cnv 5258 df-iota 5993 df-riota 6757 df-sup 8508 df-inf 8509 |
This theorem is referenced by: infxrpnf 40185 limsup0 40439 limsuppnfdlem 40446 limsup10ex 40518 |
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