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Theorem xrltletr 9717

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

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

**Proof of Theorem xrltletr**
Step	Hyp	Ref	Expression
1		simprr 522	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶)) → 𝐵 ≤ 𝐶)
2		simpl2 986	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶)) → 𝐵 ∈ ℝ^)
3		simpl3 987	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶)) → 𝐶 ∈ ℝ^)
4		xrlenlt 7944	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐵 ≤ 𝐶 ↔ ¬ 𝐶 < 𝐵))
5	2, 3, 4	syl2anc 409	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶)) → (𝐵 ≤ 𝐶 ↔ ¬ 𝐶 < 𝐵))
6	1, 5	mpbid 146	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶)) → ¬ 𝐶 < 𝐵)
7		simprl 521	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶)) → 𝐴 < 𝐵)
8		xrltso 9709	. . . . . 6 ⊢ < Or ℝ^*
9		sowlin 4282	. . . . . 6 ⊢ (( < Or ℝ^* ∧ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*)) → (𝐴 < 𝐵 → (𝐴 < 𝐶 ∨ 𝐶 < 𝐵)))
10	8, 9	mpan 421	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 < 𝐵 → (𝐴 < 𝐶 ∨ 𝐶 < 𝐵)))
11	10	adantr 274	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶)) → (𝐴 < 𝐵 → (𝐴 < 𝐶 ∨ 𝐶 < 𝐵)))
12	7, 11	mpd 13	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶)) → (𝐴 < 𝐶 ∨ 𝐶 < 𝐵))
13	6, 12	ecased 1331	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶)) → 𝐴 < 𝐶)
14	13	ex 114	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 103 ↔ wb 104 ∨ wo 698 ∧ w3a 963 ∈ wcel 2128 class class class wbr 3967 Or wor 4257 ℝ^*cxr 7913 < clt 7914 ≤ cle 7915

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4084 ax-pow 4137 ax-pr 4171 ax-un 4395 ax-setind 4498 ax-cnex 7825 ax-resscn 7826 ax-pre-ltirr 7846 ax-pre-ltwlin 7847 ax-pre-lttrn 7848

This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-br 3968 df-opab 4028 df-po 4258 df-iso 4259 df-xp 4594 df-cnv 4596 df-pnf 7916 df-mnf 7917 df-xr 7918 df-ltxr 7919 df-le 7920

This theorem is referenced by: xrltletrd 9721 xrre2 9731 xrre3 9732 ge0gtmnf 9733 iooss2 9827 iccssioo 9852 icossico 9853 icossioo 9874 ioossioo 9875 ioc0 10171

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