| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 11326 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) | |
| 4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∈ wcel 2108 class class class wbr 5143 ℝ*cxr 11294 < clt 11295 ≤ cle 11296 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-cnv 5693 df-le 11301 |
| This theorem is referenced by: xrnltled 11329 supxrleub 13368 infxrgelb 13377 ixxub 13408 ixxlb 13409 icodisj 13516 supicclub2 13544 bldisj 24408 icombl 25599 ioorcl2 25607 ply1divmo 26175 ig1peu 26214 psercnlem1 26469 infxrge0gelb 32770 supxrgere 45344 supxrgelem 45348 lenelioc 45549 iccdificc 45552 limsupub 45719 fge0iccico 46385 sge0sn 46394 sge0rpcpnf 46436 pimltmnf2f 46712 pimconstlt0 46716 pimgtpnf2f 46720 pimdecfgtioo 46732 pimincfltioo 46733 |
| Copyright terms: Public domain | W3C validator |