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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 12550 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴))) | |
6 | 3, 4, 5 | syl2anc 586 | . 2 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴))) |
7 | 1, 2, 6 | mpbir2and 711 | 1 ⊢ (𝜑 → 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 class class class wbr 5068 ℝ*cxr 10676 ≤ cle 10678 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-pre-lttri 10613 ax-pre-lttrn 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 |
This theorem is referenced by: supxrre 12723 infxrre 12732 ixxub 12762 ixxlb 12763 pcadd2 16228 psmetsym 22922 xmetsym 22959 imasdsf1olem 22985 ovolunnul 24103 ovolicc 24126 voliunlem3 24155 uniioovol 24182 uniiccvol 24183 ismbfd 24242 mbflimsup 24269 itg2itg1 24339 itg2seq 24345 itg2eqa 24348 itg2split 24352 itg2mono 24356 deg1add 24699 deg1mul2 24710 deg1tm 24714 xrgepnfd 41606 supxrge 41613 infxrpnf 41728 eliccnelico 41812 liminfgelimsup 42070 liminfgelimsupuz 42076 liminflimsupclim 42095 xlimliminflimsup 42150 ismbl4 42285 rrxsnicc 42592 sge0fsum 42676 sge0split 42698 sge0iunmptlemre 42704 sge0isum 42716 sge0xaddlem2 42723 sge0reuz 42736 meale0eq0 42767 carageniuncl 42812 caratheodorylem2 42816 caragenel2d 42821 omess0 42823 ovn0lem 42854 hoidmv1lelem2 42881 hoidmv1lelem3 42882 hoidmvlelem4 42887 ovnhoi 42892 ovolval2lem 42932 ovolval5lem3 42943 |
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