xrltletrd - Intuitionistic Logic Explorer

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Theorem xrltletrd 10090

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

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

**Proof of Theorem xrltletrd**
Step	Hyp	Ref	Expression
1		xrltletrd.4	. 2 ⊢ (𝜑 → 𝐴 < 𝐵)
2		xrltletrd.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		xrltletr 10086	. . 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 2202 class class class wbr 4093 ℝ^*cxr 8255 < clt 8256 ≤ cle 8257

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 2204 ax-14 2205 ax-ext 2213 ax-sep 4212 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-cnex 8166 ax-resscn 8167 ax-pre-ltirr 8187 ax-pre-ltwlin 8188 ax-pre-lttrn 8189

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 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-nel 2499 df-ral 2516 df-rex 2517 df-rab 2520 df-v 2805 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-br 4094 df-opab 4156 df-po 4399 df-iso 4400 df-xp 4737 df-cnv 4739 df-pnf 8258 df-mnf 8259 df-xr 8260 df-ltxr 8261 df-le 8262

This theorem is referenced by: xlt2add 10159 elico2 10216 elicc2 10217 elicore 10572 xrltmaxsup 11880 xblss2ps 15198 xblss2 15199 tgioo 15348