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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 10423 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) | |
4 | 1, 2, 3 | syl2anc 581 | 1 ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∈ wcel 2166 class class class wbr 4874 ℝ*cxr 10391 < clt 10392 ≤ cle 10393 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pr 5128 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ral 3123 df-rex 3124 df-rab 3127 df-v 3417 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-br 4875 df-opab 4937 df-xp 5349 df-cnv 5351 df-le 10398 |
This theorem is referenced by: infxrgelb 12454 ixxlb 12486 infxrge0gelb 30079 supxrgere 40347 supxrgelem 40351 lenelioc 40559 iccdificc 40562 limsupub 40732 fge0iccico 41379 sge0sn 41388 sge0rpcpnf 41430 pimltmnf2 41706 pimconstlt0 41709 pimgtpnf2 41712 pimdecfgtioo 41722 pimincfltioo 41723 |
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