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Theorem xrlelttrd 10143

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 10139	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
7	3, 4, 5, 6	syl3anc 1274	. 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 2203 class class class wbr 4109 ℝ^*cxr 8307 < clt 8308 ≤ cle 8309

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-13 2205 ax-14 2206 ax-ext 2214 ax-sep 4228 ax-pow 4287 ax-pr 4322 ax-un 4554 ax-setind 4659 ax-cnex 8218 ax-resscn 8219 ax-pre-ltirr 8239 ax-pre-ltwlin 8240 ax-pre-lttrn 8241

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 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-rab 2529 df-v 2815 df-dif 3213 df-un 3215 df-in 3217 df-ss 3224 df-pw 3671 df-sn 3695 df-pr 3696 df-op 3698 df-uni 3915 df-br 4110 df-opab 4172 df-po 4417 df-iso 4418 df-xp 4755 df-cnv 4757 df-pnf 8310 df-mnf 8311 df-xr 8312 df-ltxr 8313 df-le 8314

This theorem is referenced by: xlt2add 10213 elioc2 10269 elicc2 10271 xrmaxltsup 11943 blgt0 15267 xblss2ps 15269 xblss2 15270 tgioo 15419