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Mirrors > Home > MPE Home > Th. List > xrleidd | Structured version Visualization version GIF version |
Description: 'Less than or equal to' is reflexive for extended reals. Deduction form of xrleid 12294. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
xrleidd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
Ref | Expression |
---|---|
xrleidd | ⊢ (𝜑 → 𝐴 ≤ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrleidd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
2 | xrleid 12294 | . 2 ⊢ (𝐴 ∈ ℝ* → 𝐴 ≤ 𝐴) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → 𝐴 ≤ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 class class class wbr 4886 ℝ*cxr 10410 ≤ cle 10412 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-pre-lttri 10346 ax-pre-lttrn 10347 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-po 5274 df-so 5275 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 |
This theorem is referenced by: xleadd1a 12395 supxrre 12469 infxrre 12478 ioounsn 12613 snunioo 12615 snunico 12616 limsupgre 14620 limsupbnd1 14621 limsupbnd2 14622 pcdvdstr 15984 pcadd 15997 imasdsf1olem 22586 blssps 22637 blss 22638 blcld 22718 nmolb 22929 metds0 23061 metdstri 23062 metdseq0 23065 itg2eqa 23949 mdeglt 24262 deg1lt 24294 eliccelico 30103 elicoelioo 30104 difioo 30108 xrge0omnd 30273 esumpmono 30739 signsply0 31228 iocinico 38755 xadd0ge 40444 infxrpnf 40580 monoordxrv 40617 iooiinioc 40691 icossico2 40699 limcresiooub 40782 liminflelimsupuz 40925 ismbl4 41137 sge0prle 41542 iunhoiioo 41817 iccpartleu 42396 iccpartgel 42397 |
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