iccdisj - Mathbox for Zhi Wang

	Mathbox for Zhi Wang		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > iccdisj		Structured version Visualization version GIF version

Theorem iccdisj 48796

Description: If the upper bound of one closed interval is less than the lower bound of the other, the intervals are disjoint. (Contributed by Zhi Wang, 9-Sep-2024.)

**Assertion**
Ref	Expression
iccdisj	⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^*)) ∧ 𝐵 < 𝐶) → ((𝐴[,]𝐵) ∩ (𝐶[,]𝐷)) = ∅)

**Proof of Theorem iccdisj**
Step	Hyp	Ref	Expression
1		simplll 775	. 2 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^)) ∧ 𝐵 < 𝐶) → 𝐴 ∈ ℝ^)
2		simplrr 778	. 2 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^)) ∧ 𝐵 < 𝐶) → 𝐷 ∈ ℝ^)
3		simpr 484	. 2 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^*)) ∧ 𝐵 < 𝐶) → 𝐵 < 𝐶)
4		iccdisj2 48795	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^* ∧ 𝐵 < 𝐶) → ((𝐴[,]𝐵) ∩ (𝐶[,]𝐷)) = ∅)
5	1, 2, 3, 4	syl3anc 1373	1 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^*)) ∧ 𝐵 < 𝐶) → ((𝐴[,]𝐵) ∩ (𝐶[,]𝐷)) = ∅)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∩ cin 3950 ∅c0 4333 class class class wbr 5143 (class class class)co 7431 ℝ^*cxr 11294 < clt 11295 [,]cicc 13390

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-pre-lttri 11229 ax-pre-lttrn 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-ico 13393 df-icc 13394

This theorem is referenced by: (None)

W3C validator