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Theorem xrltletr 10164

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 533	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶)) → 𝐵 ≤ 𝐶)
2		simpl2 1028	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶)) → 𝐵 ∈ ℝ^)
3		simpl3 1029	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶)) → 𝐶 ∈ ℝ^)
4		xrlenlt 8356	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐵 ≤ 𝐶 ↔ ¬ 𝐶 < 𝐵))
5	2, 3, 4	syl2anc 411	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶)) → (𝐵 ≤ 𝐶 ↔ ¬ 𝐶 < 𝐵))
6	1, 5	mpbid 147	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶)) → ¬ 𝐶 < 𝐵)
7		simprl 531	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶)) → 𝐴 < 𝐵)
8		xrltso 10153	. . . . . 6 ⊢ < Or ℝ^*
9		sowlin 4447	. . . . . 6 ⊢ (( < Or ℝ^* ∧ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*)) → (𝐴 < 𝐵 → (𝐴 < 𝐶 ∨ 𝐶 < 𝐵)))
10	8, 9	mpan 424	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 < 𝐵 → (𝐴 < 𝐶 ∨ 𝐶 < 𝐵)))
11	10	adantr 276	. . . 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 1386	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶)) → 𝐴 < 𝐶)
14	13	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 716 ∧ w3a 1005 ∈ wcel 2205 class class class wbr 4115 Or wor 4422 ℝ^*cxr 8325 < clt 8326 ≤ cle 8327

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 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-14 2208 ax-ext 2216 ax-sep 4234 ax-pow 4293 ax-pr 4328 ax-un 4560 ax-setind 4666 ax-cnex 8236 ax-resscn 8237 ax-pre-ltirr 8257 ax-pre-ltwlin 8258 ax-pre-lttrn 8259

This theorem depends on definitions: df-bi 117 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-rab 2531 df-v 2817 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-pw 3677 df-sn 3701 df-pr 3702 df-op 3704 df-uni 3921 df-br 4116 df-opab 4178 df-po 4423 df-iso 4424 df-xp 4762 df-cnv 4764 df-pnf 8328 df-mnf 8329 df-xr 8330 df-ltxr 8331 df-le 8332

This theorem is referenced by: xrltletrd 10168 xrre2 10178 xrre3 10179 ge0gtmnf 10180 iooss2 10274 iccssioo 10299 icossico 10300 icossioo 10321 ioossioo 10322 ioc0 10651

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