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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 12535 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴))) | |
6 | 3, 4, 5 | syl2anc 587 | . 2 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴))) |
7 | 1, 2, 6 | mpbir2and 712 | 1 ⊢ (𝜑 → 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 class class class wbr 5030 ℝ*cxr 10663 ≤ cle 10665 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-pre-lttri 10600 ax-pre-lttrn 10601 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 |
This theorem is referenced by: supxrre 12708 infxrre 12717 ixxub 12747 ixxlb 12748 pcadd2 16216 psmetsym 22917 xmetsym 22954 imasdsf1olem 22980 ovolunnul 24104 ovolicc 24127 voliunlem3 24156 uniioovol 24183 uniiccvol 24184 ismbfd 24243 mbflimsup 24270 itg2itg1 24340 itg2seq 24346 itg2eqa 24349 itg2split 24353 itg2mono 24357 deg1add 24704 deg1mul2 24715 deg1tm 24719 xrgepnfd 41963 supxrge 41970 infxrpnf 42084 eliccnelico 42166 liminfgelimsup 42424 liminfgelimsupuz 42430 liminflimsupclim 42449 xlimliminflimsup 42504 ismbl4 42635 rrxsnicc 42942 sge0fsum 43026 sge0split 43048 sge0iunmptlemre 43054 sge0isum 43066 sge0xaddlem2 43073 sge0reuz 43086 meale0eq0 43117 carageniuncl 43162 caratheodorylem2 43166 caragenel2d 43171 omess0 43173 ovn0lem 43204 hoidmv1lelem2 43231 hoidmv1lelem3 43232 hoidmvlelem4 43237 ovnhoi 43242 ovolval2lem 43282 ovolval5lem3 43293 |
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