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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 4573 | . . . 4 ⊢ {𝐵} = {𝐵, 𝐵} | |
2 | 1 | infeq1i 8936 | . . 3 ⊢ inf({𝐵}, 𝐴, 𝑅) = inf({𝐵, 𝐵}, 𝐴, 𝑅) |
3 | infpr 8961 | . . . 4 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴) → inf({𝐵, 𝐵}, 𝐴, 𝑅) = if(𝐵𝑅𝐵, 𝐵, 𝐵)) | |
4 | 3 | 3anidm23 1417 | . . 3 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → inf({𝐵, 𝐵}, 𝐴, 𝑅) = if(𝐵𝑅𝐵, 𝐵, 𝐵)) |
5 | 2, 4 | syl5eq 2868 | . 2 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → inf({𝐵}, 𝐴, 𝑅) = if(𝐵𝑅𝐵, 𝐵, 𝐵)) |
6 | ifid 4505 | . 2 ⊢ if(𝐵𝑅𝐵, 𝐵, 𝐵) = 𝐵 | |
7 | 5, 6 | syl6eq 2872 | 1 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → inf({𝐵}, 𝐴, 𝑅) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ifcif 4466 {csn 4560 {cpr 4562 class class class wbr 5058 Or wor 5467 infcinf 8899 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-po 5468 df-so 5469 df-cnv 5557 df-iota 6308 df-riota 7108 df-sup 8900 df-inf 8901 |
This theorem is referenced by: infxrpnf 41714 limsup0 41968 limsuppnfdlem 41975 limsup10ex 42047 |
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