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Theorem xrlelttrd 10044

Description: Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015.)

**Assertion**
Ref	Expression
xrlelttrd	⊢ (𝜑 → 𝐴 < 𝐶)

**Proof of Theorem xrlelttrd**
Step	Hyp	Ref	Expression
1		xrlelttrd.4	. 2 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
2		xrlelttrd.5	. 2 ⊢ (𝜑 → 𝐵 < 𝐶)
3		xrlttrd.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
4		xrlttrd.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
5		xrlttrd.3	. . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
6		xrlelttr 10040	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
7	3, 4, 5, 6	syl3anc 1273	. 2 ⊢ (𝜑 → ((𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
8	1, 2, 7	mp2and 433	1 ⊢ (𝜑 → 𝐴 < 𝐶)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ∈ wcel 2202 class class class wbr 4088 ℝ^*cxr 8212 < clt 8213 ≤ cle 8214

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 8122 ax-resscn 8123 ax-pre-ltirr 8143 ax-pre-ltwlin 8144 ax-pre-lttrn 8145

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 8215 df-mnf 8216 df-xr 8217 df-ltxr 8218 df-le 8219

This theorem is referenced by: xlt2add 10114 elioc2 10170 elicc2 10172 xrmaxltsup 11818 blgt0 15125 xblss2ps 15127 xblss2 15128 tgioo 15277