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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 13192 | . . 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 1536 ∈ wcel 2105 class class class wbr 5147 ℝ*cxr 11291 ≤ cle 11293 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-pre-lttri 11226 ax-pre-lttrn 11227 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-po 5596 df-so 5597 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 |
This theorem is referenced by: supxrre 13365 infxrre 13374 ixxub 13404 ixxlb 13405 pcadd2 16923 psmetsym 24335 xmetsym 24372 imasdsf1olem 24398 ovolunnul 25548 ovolicc 25571 voliunlem3 25600 uniioovol 25627 uniiccvol 25628 ismbfd 25687 mbflimsup 25714 itg2itg1 25785 itg2seq 25791 itg2eqa 25794 itg2split 25798 itg2mono 25802 deg1add 26156 deg1mul2 26167 deg1tm 26172 xrgepnfd 45280 supxrge 45287 infxrpnf 45395 eliccnelico 45481 liminfgelimsup 45737 liminfgelimsupuz 45743 liminflimsupclim 45762 xlimliminflimsup 45817 ismbl4 45948 rrxsnicc 46255 sge0fsum 46342 sge0split 46364 sge0iunmptlemre 46370 sge0isum 46382 sge0xaddlem2 46389 sge0reuz 46402 meale0eq0 46433 carageniuncl 46478 caratheodorylem2 46482 caragenel2d 46487 omess0 46489 ovn0lem 46520 hoidmv1lelem2 46547 hoidmv1lelem3 46548 hoidmvlelem4 46553 ovnhoi 46558 ovolval2lem 46598 ovolval5lem3 46609 |
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