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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 10700 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) | |
4 | 1, 2, 3 | syl2anc 586 | 1 ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∈ wcel 2110 class class class wbr 5058 ℝ*cxr 10668 < clt 10669 ≤ cle 10670 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 df-opab 5121 df-xp 5555 df-cnv 5557 df-le 10675 |
This theorem is referenced by: xrnltled 10703 supxrleub 12713 infxrgelb 12722 ixxub 12753 ixxlb 12754 icodisj 12856 supicclub2 12883 bldisj 23002 icombl 24159 ioorcl2 24167 ply1divmo 24723 ig1peu 24759 psercnlem1 25007 infxrge0gelb 30484 supxrgere 41594 supxrgelem 41598 lenelioc 41805 iccdificc 41808 limsupub 41978 fge0iccico 42646 sge0sn 42655 sge0rpcpnf 42697 pimltmnf2 42973 pimconstlt0 42976 pimgtpnf2 42979 pimdecfgtioo 42989 pimincfltioo 42990 |
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