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Theorem xrletri3 13150

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 13139	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
2	1	biancomd 467	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 = 𝐵 ↔ (¬ 𝐵 < 𝐴 ∧ ¬ 𝐴 < 𝐵)))
3		xrlenlt 11241	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴))
4		xrlenlt 11241	. . . 4 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (𝐵 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐵))
5	4	ancoms 462	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐵 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐵))
6	3, 5	anbi12d 641	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴) ↔ (¬ 𝐵 < 𝐴 ∧ ¬ 𝐴 < 𝐵)))
7	2, 6	bitr4d 284	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 399 = wceq 1559 ∈ wcel 2141 class class class wbr 5097 ℝ^*cxr 11209 < clt 11210 ≤ cle 11211

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7713 ax-cnex 11123 ax-resscn 11124 ax-pre-lttri 11141 ax-pre-lttrn 11142

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-br 5098 df-opab 5160 df-mpt 5179 df-id 5538 df-po 5551 df-so 5552 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-iota 6472 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-er 8672 df-en 8922 df-dom 8923 df-sdom 8924 df-pnf 11212 df-mnf 11213 df-xr 11214 df-ltxr 11215 df-le 11216

This theorem is referenced by: xrletrid 13151 xrmaxeq 13176 xrmineq 13177 xleadd1a 13250 xsubge0 13258 xlemul1a 13285 hashle00 14407 limsupval2 15498 pc2dvds 16906 pc11 16907 letsr 18616 xrge0omnd 21485 isxmet2d 24375 xmetgt0 24406 prdsxmetlem 24416 xblss2 24450 nmo0 24783 nmoid 24790 xrsxmet 24858 ovolssnul 25537 ovolctb 25540 ovoliunnul 25557 ovolre 25575 volsup 25606 vitalilem5 25662 itg2mulc 25797 umgrislfupgrlem 29280 upgr2pthnlp 29889 xeqlelt 32939 xrstos 33149 metideq 34151 metider 34152 esumpad2 34314 esumrnmpt2 34326 measle0 34466 inelcarsg 34569 carsggect 34576 carsgclctun 34579 omsmeas 34581 ovoliunnfl 38122 volsupnfl 38125 iccintsng 46060 liminfval2 46303