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Theorem iccssico2 13373

Description: Condition for a closed interval to be a subset of a closed-below, open-above interval. (Contributed by Mario Carneiro, 20-Feb-2015.)

**Assertion**
Ref	Expression
iccssico2	⊢ ((𝐶 ∈ (𝐴[,)𝐵) ∧ 𝐷 ∈ (𝐴[,)𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴[,)𝐵))

Proof of Theorem iccssico2 Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ico 13304	. . . 4 ⊢ [,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
2	1	elmpocl1 7609	. . 3 ⊢ (𝐶 ∈ (𝐴[,)𝐵) → 𝐴 ∈ ℝ^*)
3	2	adantr 480	. 2 ⊢ ((𝐶 ∈ (𝐴[,)𝐵) ∧ 𝐷 ∈ (𝐴[,)𝐵)) → 𝐴 ∈ ℝ^*)
4	1	elmpocl2 7610	. . 3 ⊢ (𝐶 ∈ (𝐴[,)𝐵) → 𝐵 ∈ ℝ^*)
5	4	adantr 480	. 2 ⊢ ((𝐶 ∈ (𝐴[,)𝐵) ∧ 𝐷 ∈ (𝐴[,)𝐵)) → 𝐵 ∈ ℝ^*)
6	1	elixx3g 13311	. . . . 5 ⊢ (𝐶 ∈ (𝐴[,)𝐵) ↔ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)))
7	6	simprbi 497	. . . 4 ⊢ (𝐶 ∈ (𝐴[,)𝐵) → (𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵))
8	7	simpld 494	. . 3 ⊢ (𝐶 ∈ (𝐴[,)𝐵) → 𝐴 ≤ 𝐶)
9	8	adantr 480	. 2 ⊢ ((𝐶 ∈ (𝐴[,)𝐵) ∧ 𝐷 ∈ (𝐴[,)𝐵)) → 𝐴 ≤ 𝐶)
10	1	elixx3g 13311	. . . . 5 ⊢ (𝐷 ∈ (𝐴[,)𝐵) ↔ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐷 ∧ 𝐷 < 𝐵)))
11	10	simprbi 497	. . . 4 ⊢ (𝐷 ∈ (𝐴[,)𝐵) → (𝐴 ≤ 𝐷 ∧ 𝐷 < 𝐵))
12	11	simprd 495	. . 3 ⊢ (𝐷 ∈ (𝐴[,)𝐵) → 𝐷 < 𝐵)
13	12	adantl 481	. 2 ⊢ ((𝐶 ∈ (𝐴[,)𝐵) ∧ 𝐷 ∈ (𝐴[,)𝐵)) → 𝐷 < 𝐵)
14		iccssico 13371	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐶 ∧ 𝐷 < 𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴[,)𝐵))
15	3, 5, 9, 13, 14	syl22anc 839	1 ⊢ ((𝐶 ∈ (𝐴[,)𝐵) ∧ 𝐷 ∈ (𝐴[,)𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴[,)𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 ∈ wcel 2114 {crab 3389 ⊆ wss 3889 class class class wbr 5085 (class class class)co 7367 ℝ^*cxr 11178 < clt 11179 ≤ cle 11180 [,)cico 13300 [,]cicc 13301

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-pre-lttri 11112 ax-pre-lttrn 11113

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-id 5526 df-po 5539 df-so 5540 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-ov 7370 df-oprab 7371 df-mpo 7372 df-1st 7942 df-2nd 7943 df-er 8643 df-en 8894 df-dom 8895 df-sdom 8896 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-ico 13304 df-icc 13305

This theorem is referenced by: icopnfhmeo 24910

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