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Theorem xrletri3 10206

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 10199	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 = 𝐵 ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴)))
2		ancom 266	. . 3 ⊢ ((¬ 𝐵 < 𝐴 ∧ ¬ 𝐴 < 𝐵) ↔ (¬ 𝐴 < 𝐵 ∧ ¬ 𝐵 < 𝐴))
3	1, 2	bitr4di 198	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 = 𝐵 ↔ (¬ 𝐵 < 𝐴 ∧ ¬ 𝐴 < 𝐵)))
4		xrlenlt 8390	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴))
5		xrlenlt 8390	. . . 4 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (𝐵 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐵))
6	5	ancoms 268	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐵 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐵))
7	4, 6	anbi12d 477	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴) ↔ (¬ 𝐵 < 𝐴 ∧ ¬ 𝐴 < 𝐵)))
8	3, 7	bitr4d 191	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴)))

Colors of variables: wff set class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1402 ∈ wcel 2209 class class class wbr 4130 ℝ^*cxr 8359 < clt 8360 ≤ cle 8361

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4249 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-cnex 8270 ax-resscn 8271 ax-pre-ltirr 8291 ax-pre-apti 8294

This proof depends on definitions: df-bi 117 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-rab 2537 df-v 2823 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-br 4131 df-opab 4193 df-xp 4780 df-cnv 4782 df-pnf 8362 df-mnf 8363 df-xr 8364 df-ltxr 8365 df-le 8366

This theorem is used by: xrletrid 10207 xleadd1a 10275 xsubge0 10283 pc2dvds 13109 pc11 13110 psmetsym 15430 isxmet2d 15449 xmetsym 15469 xblss2 15506 xmetxp 15608

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