xrltletr - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > xrltletr		GIF version

Theorem xrltletr 8877

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 498	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶)) → 𝐵 ≤ 𝐶)
2		simpl2 942	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶)) → 𝐵 ∈ ℝ^)
3		simpl3 943	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶)) → 𝐶 ∈ ℝ^)
4		xrlenlt 7177	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐵 ≤ 𝐶 ↔ ¬ 𝐶 < 𝐵))
5	2, 3, 4	syl2anc 403	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶)) → (𝐵 ≤ 𝐶 ↔ ¬ 𝐶 < 𝐵))
6	1, 5	mpbid 145	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶)) → ¬ 𝐶 < 𝐵)
7		simprl 497	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶)) → 𝐴 < 𝐵)
8		xrltso 8871	. . . . . 6 ⊢ < Or ℝ^*
9		sowlin 4075	. . . . . 6 ⊢ (( < Or ℝ^* ∧ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*)) → (𝐴 < 𝐵 → (𝐴 < 𝐶 ∨ 𝐶 < 𝐵)))
10	8, 9	mpan 414	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 < 𝐵 → (𝐴 < 𝐶 ∨ 𝐶 < 𝐵)))
11	10	adantr 270	. . . 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 1280	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶)) → 𝐴 < 𝐶)
14	13	ex 113	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 102 ↔ wb 103 ∨ wo 661 ∧ w3a 919 ∈ wcel 1433 class class class wbr 3785 Or wor 4050 ℝ^*cxr 7152 < clt 7153 ≤ cle 7154

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090

This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-po 4051 df-iso 4052 df-xp 4369 df-cnv 4371 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159

This theorem is referenced by: xrltletrd 8881 xrre2 8888 xrre3 8889 ge0gtmnf 8890 iooss2 8940 iccssioo 8965 icossico 8966 icossioo 8987 ioossioo 8988 ioc0 9271

W3C validator