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Theorem xrlelttr 10210

Description: Transitive law for ordering on extended reals. (Contributed by NM, 19-Jan-2006.)

**Assertion**
Ref	Expression
xrlelttr	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))

**Proof of Theorem xrlelttr**
Step	Hyp	Ref	Expression
1		simprl 535	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶)) → 𝐴 ≤ 𝐵)
2		simpl1 1031	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶)) → 𝐴 ∈ ℝ^)
3		simpl2 1032	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶)) → 𝐵 ∈ ℝ^)
4		xrlenlt 8390	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴))
5	2, 3, 4	syl2anc 415	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶)) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴))
6	1, 5	mpbid 147	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶)) → ¬ 𝐵 < 𝐴)
7	6	pm2.21d 628	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶)) → (𝐵 < 𝐴 → 𝐴 < 𝐶))
8		idd 21	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶)) → (𝐴 < 𝐶 → 𝐴 < 𝐶))
9		simprr 537	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶)) → 𝐵 < 𝐶)
10		simpl3 1033	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶)) → 𝐶 ∈ ℝ^)
11		xrltso 10200	. . . . . 6 ⊢ < Or ℝ^*
12		sowlin 4465	. . . . . 6 ⊢ (( < Or ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*)) → (𝐵 < 𝐶 → (𝐵 < 𝐴 ∨ 𝐴 < 𝐶)))
13	11, 12	mpan 428	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (𝐵 < 𝐶 → (𝐵 < 𝐴 ∨ 𝐴 < 𝐶)))
14	3, 10, 2, 13	syl3anc 1278	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶)) → (𝐵 < 𝐶 → (𝐵 < 𝐴 ∨ 𝐴 < 𝐶)))
15	9, 14	mpd 13	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶)) → (𝐵 < 𝐴 ∨ 𝐴 < 𝐶))
16	7, 8, 15	mpjaod 730	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶)) → 𝐴 < 𝐶)
17	16	ex 115	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))

Colors of variables: wff set class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 720 ∧ w3a 1009 ∈ wcel 2209 class class class wbr 4130 Or wor 4440 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-ltwlin 8292 ax-pre-lttrn 8293

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-po 4441 df-iso 4442 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: xrlelttrd 10214 xrre 10224 xrre2 10225 iooss1 10320 iccssioo 10346 iccssico 10349 iocssioo 10367 ioossioo 10369 ico0 10698 bldisj 15504 xblm 15520 blsscls2 15596 metcnpi3 15620

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