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Theorem xrletri3 13048

Description: Trichotomy law for extended reals. (Contributed by FL, 2-Aug-2009.)

**Assertion**
Ref	Expression
xrletri3	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴)))

**Proof of Theorem xrletri3**
Step	Hyp	Ref	Expression
1		xrlttri3 13037	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
2	1	biancomd 463	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 = 𝐵 ↔ (¬ 𝐵 < 𝐴 ∧ ¬ 𝐴 < 𝐵)))
3		xrlenlt 11172	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴))
4		xrlenlt 11172	. . . 4 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (𝐵 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐵))
5	4	ancoms 458	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐵 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐵))
6	3, 5	anbi12d 632	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴) ↔ (¬ 𝐵 < 𝐴 ∧ ¬ 𝐴 < 𝐵)))
7	2, 6	bitr4d 282	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2111 class class class wbr 5086 ℝ^*cxr 11140 < clt 11141 ≤ cle 11142

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 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5229 ax-nul 5239 ax-pow 5298 ax-pr 5365 ax-un 7663 ax-cnex 11057 ax-resscn 11058 ax-pre-lttri 11075 ax-pre-lttrn 11076

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4279 df-if 4471 df-pw 4547 df-sn 4572 df-pr 4574 df-op 4578 df-uni 4855 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5506 df-po 5519 df-so 5520 df-xp 5617 df-rel 5618 df-cnv 5619 df-co 5620 df-dm 5621 df-rn 5622 df-res 5623 df-ima 5624 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-er 8617 df-en 8865 df-dom 8866 df-sdom 8867 df-pnf 11143 df-mnf 11144 df-xr 11145 df-ltxr 11146 df-le 11147

This theorem is referenced by: xrletrid 13049 xrmaxeq 13073 xrmineq 13074 xleadd1a 13147 xsubge0 13155 xlemul1a 13182 hashle00 14302 limsupval2 15382 pc2dvds 16786 pc11 16787 letsr 18494 xrge0omnd 21377 isxmet2d 24237 xmetgt0 24268 prdsxmetlem 24278 xblss2 24312 nmo0 24645 nmoid 24652 xrsxmet 24720 ovolssnul 25410 ovolctb 25413 ovoliunnul 25430 ovolre 25448 volsup 25479 vitalilem5 25535 itg2mulc 25670 umgrislfupgrlem 29095 upgr2pthnlp 29705 xeqlelt 32751 xrstos 32983 metideq 33898 metider 33899 esumpad2 34061 esumrnmpt2 34073 measle0 34213 inelcarsg 34316 carsggect 34323 carsgclctun 34326 omsmeas 34328 ovoliunnfl 37702 volsupnfl 37705 iccintsng 45563 liminfval2 45806