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Mirrors > Home > MPE Home > Th. List > xrltle | Structured version Visualization version GIF version |
Description: 'Less than' implies 'less than or equal' for extended reals. (Contributed by NM, 19-Jan-2006.) |
Ref | Expression |
---|---|
xrltle | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 → 𝐴 ≤ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | orc 861 | . 2 ⊢ (𝐴 < 𝐵 → (𝐴 < 𝐵 ∨ 𝐴 = 𝐵)) | |
2 | xrleloe 12527 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ≤ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵))) | |
3 | 1, 2 | syl5ibr 247 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 → 𝐴 ≤ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∨ wo 841 = wceq 1528 ∈ wcel 2105 class class class wbr 5058 ℝ*cxr 10663 < clt 10664 ≤ cle 10665 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-cnex 10582 ax-resscn 10583 ax-pre-lttri 10600 ax-pre-lttrn 10601 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4833 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-er 8279 df-en 8499 df-dom 8500 df-sdom 8501 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 |
This theorem is referenced by: xrltled 12533 xrletri 12536 xrletr 12541 qextltlem 12585 xmulge0 12667 supxrunb1 12702 ico0 12774 ioc0 12775 ioossicc 12812 icossicc 12814 iocssicc 12815 ioossico 12816 snunioo 12854 snunico 12855 ioopnfsup 13222 icopnfsup 13223 hashnnn0genn0 13693 leordtval2 21750 lecldbas 21757 blcls 23045 stdbdxmet 23054 stdbdmopn 23057 metcnpi3 23085 xrsmopn 23349 metnrmlem1a 23395 bndth 23491 ovolgelb 24010 icombl 24094 ioorf 24103 ioorinv2 24105 itg2seq 24272 tanord1 25048 dvloglem 25158 iocinif 30431 esumpinfsum 31236 omssubadd 31458 elicc3 33563 tan2h 34766 heicant 34809 itg2addnclem 34825 radcnvrat 40526 ioossioc 41646 ioossioobi 41673 fouriersw 42397 iccpartnel 43445 |
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