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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 866 | . 2 ⊢ (𝐴 < 𝐵 → (𝐴 < 𝐵 ∨ 𝐴 = 𝐵)) | |
2 | xrleloe 13110 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ≤ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵))) | |
3 | 1, 2 | imbitrrid 245 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 → 𝐴 ≤ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 class class class wbr 5144 ℝ*cxr 11234 < clt 11235 ≤ cle 11236 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-cnex 11153 ax-resscn 11154 ax-pre-lttri 11171 ax-pre-lttrn 11172 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-er 8691 df-en 8928 df-dom 8929 df-sdom 8930 df-pnf 11237 df-mnf 11238 df-xr 11239 df-ltxr 11240 df-le 11241 |
This theorem is referenced by: xrltled 13116 xrletri 13119 xrletr 13124 qextltlem 13168 xmulge0 13250 supxrunb1 13285 ico0 13357 ioc0 13358 ioossicc 13397 icossicc 13400 iocssicc 13401 ioossico 13402 snunioo 13442 snunico 13443 ioopnfsup 13816 icopnfsup 13817 hashnnn0genn0 14290 leordtval2 22685 lecldbas 22692 blcls 23984 stdbdxmet 23993 stdbdmopn 23996 metcnpi3 24024 xrsmopn 24297 metnrmlem1a 24343 bndth 24443 ovolgelb 24966 icombl 25050 ioorf 25059 ioorinv2 25061 itg2seq 25229 tanord1 26015 dvloglem 26125 iocinif 31963 esumpinfsum 33006 omssubadd 33230 elicc3 35107 tan2h 36385 heicant 36428 itg2addnclem 36444 radcnvrat 42944 ioossioc 44078 ioossioobi 44103 fouriersw 44820 iccpartnel 45979 i0oii 47392 io1ii 47393 |
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