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Theorem xrlelttr 10041

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 531	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶)) → 𝐴 ≤ 𝐵)
2		simpl1 1026	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶)) → 𝐴 ∈ ℝ^)
3		simpl2 1027	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶)) → 𝐵 ∈ ℝ^)
4		xrlenlt 8244	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴))
5	2, 3, 4	syl2anc 411	. . . . 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 624	. . 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 533	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶)) → 𝐵 < 𝐶)
10		simpl3 1028	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶)) → 𝐶 ∈ ℝ^)
11		xrltso 10031	. . . . . 6 ⊢ < Or ℝ^*
12		sowlin 4417	. . . . . 6 ⊢ (( < Or ℝ^* ∧ (𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*)) → (𝐵 < 𝐶 → (𝐵 < 𝐴 ∨ 𝐴 < 𝐶)))
13	11, 12	mpan 424	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (𝐵 < 𝐶 → (𝐵 < 𝐴 ∨ 𝐴 < 𝐶)))
14	3, 10, 2, 13	syl3anc 1273	. . . 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 725	. 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

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 715 ∧ w3a 1004 ∈ wcel 2202 class class class wbr 4088 Or wor 4392 ℝ^*cxr 8213 < clt 8214 ≤ cle 8215

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-sep 4207 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-cnex 8123 ax-resscn 8124 ax-pre-ltirr 8144 ax-pre-ltwlin 8145 ax-pre-lttrn 8146

This theorem depends on definitions: df-bi 117 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-rab 2519 df-v 2804 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-br 4089 df-opab 4151 df-po 4393 df-iso 4394 df-xp 4731 df-cnv 4733 df-pnf 8216 df-mnf 8217 df-xr 8218 df-ltxr 8219 df-le 8220

This theorem is referenced by: xrlelttrd 10045 xrre 10055 xrre2 10056 iooss1 10151 iccssioo 10177 iccssico 10180 iocssioo 10198 ioossioo 10200 ico0 10522 bldisj 15131 xblm 15147 blsscls2 15223 metcnpi3 15247

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