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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 863 | . 2 ⊢ (𝐴 < 𝐵 → (𝐴 < 𝐵 ∨ 𝐴 = 𝐵)) | |
2 | xrleloe 12540 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ≤ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵))) | |
3 | 1, 2 | syl5ibr 248 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 → 𝐴 ≤ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 class class class wbr 5068 ℝ*cxr 10676 < clt 10677 ≤ cle 10678 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-pre-lttri 10613 ax-pre-lttrn 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 |
This theorem is referenced by: xrltled 12546 xrletri 12549 xrletr 12554 qextltlem 12598 xmulge0 12680 supxrunb1 12715 ico0 12787 ioc0 12788 ioossicc 12825 icossicc 12827 iocssicc 12828 ioossico 12829 snunioo 12867 snunico 12868 ioopnfsup 13235 icopnfsup 13236 hashnnn0genn0 13706 leordtval2 21822 lecldbas 21829 blcls 23118 stdbdxmet 23127 stdbdmopn 23130 metcnpi3 23158 xrsmopn 23422 metnrmlem1a 23468 bndth 23564 ovolgelb 24083 icombl 24167 ioorf 24176 ioorinv2 24178 itg2seq 24345 tanord1 25123 dvloglem 25233 iocinif 30506 esumpinfsum 31338 omssubadd 31560 elicc3 33667 tan2h 34886 heicant 34929 itg2addnclem 34945 radcnvrat 40653 ioossioc 41773 ioossioobi 41800 fouriersw 42523 iccpartnel 43605 |
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