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| Mirrors > Home > MPE Home > Th. List > xrletrid | Structured version Visualization version GIF version | ||
| Description: Trichotomy law for extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| Ref | Expression |
|---|---|
| xrletrid.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
| xrletrid.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
| xrletrid.3 | ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
| xrletrid.4 | ⊢ (𝜑 → 𝐵 ≤ 𝐴) |
| Ref | Expression |
|---|---|
| xrletrid | ⊢ (𝜑 → 𝐴 = 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | xrletrid.3 | . 2 ⊢ (𝜑 → 𝐴 ≤ 𝐵) | |
| 2 | xrletrid.4 | . 2 ⊢ (𝜑 → 𝐵 ≤ 𝐴) | |
| 3 | xrletrid.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
| 4 | xrletrid.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
| 5 | xrletri3 13196 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴))) | |
| 6 | 3, 4, 5 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴))) |
| 7 | 1, 2, 6 | mpbir2and 713 | 1 ⊢ (𝜑 → 𝐴 = 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 ℝ*cxr 11294 ≤ 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-pre-lttri 11229 ax-pre-lttrn 11230 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 |
| This theorem is referenced by: supxrre 13369 infxrre 13378 ixxub 13408 ixxlb 13409 pcadd2 16928 psmetsym 24320 xmetsym 24357 imasdsf1olem 24383 ovolunnul 25535 ovolicc 25558 voliunlem3 25587 uniioovol 25614 uniiccvol 25615 ismbfd 25674 mbflimsup 25701 itg2itg1 25771 itg2seq 25777 itg2eqa 25780 itg2split 25784 itg2mono 25788 deg1add 26142 deg1mul2 26153 deg1tm 26158 xrgepnfd 45342 supxrge 45349 infxrpnf 45457 eliccnelico 45542 liminfgelimsup 45797 liminfgelimsupuz 45803 liminflimsupclim 45822 xlimliminflimsup 45877 ismbl4 46008 rrxsnicc 46315 sge0fsum 46402 sge0split 46424 sge0iunmptlemre 46430 sge0isum 46442 sge0xaddlem2 46449 sge0reuz 46462 meale0eq0 46493 carageniuncl 46538 caratheodorylem2 46542 caragenel2d 46547 omess0 46549 ovn0lem 46580 hoidmv1lelem2 46607 hoidmv1lelem3 46608 hoidmvlelem4 46613 ovnhoi 46618 ovolval2lem 46658 ovolval5lem3 46669 |
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