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Mirrors > Home > MPE Home > Th. List > xrlenltd | Structured version Visualization version GIF version |
Description: "Less than or equal to" expressed in terms of "less than", for extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
xrlenltd.a | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
xrlenltd.b | ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
Ref | Expression |
---|---|
xrlenltd | ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrlenltd.a | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
2 | xrlenltd.b | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
3 | xrlenlt 11311 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) | |
4 | 1, 2, 3 | syl2anc 582 | 1 ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∈ wcel 2098 class class class wbr 5149 ℝ*cxr 11279 < clt 11280 ≤ cle 11281 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5150 df-opab 5212 df-xp 5684 df-cnv 5686 df-le 11286 |
This theorem is referenced by: xrnltled 11314 supxrleub 13340 infxrgelb 13349 ixxub 13380 ixxlb 13381 icodisj 13488 supicclub2 13516 bldisj 24348 icombl 25537 ioorcl2 25545 ply1divmo 26116 ig1peu 26154 psercnlem1 26407 infxrge0gelb 32618 supxrgere 44853 supxrgelem 44857 lenelioc 45059 iccdificc 45062 limsupub 45230 fge0iccico 45896 sge0sn 45905 sge0rpcpnf 45947 pimltmnf2f 46223 pimconstlt0 46227 pimgtpnf2f 46231 pimdecfgtioo 46243 pimincfltioo 46244 |
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