iccssioo2 - Metamath Proof Explorer

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Theorem iccssioo2 13482

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

**Assertion**
Ref	Expression
iccssioo2	⊢ ((𝐶 ∈ (𝐴(,)𝐵) ∧ 𝐷 ∈ (𝐴(,)𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴(,)𝐵))

**Proof of Theorem iccssioo2**
Step	Hyp	Ref	Expression
1		ne0i 4364	. . . 4 ⊢ (𝐶 ∈ (𝐴(,)𝐵) → (𝐴(,)𝐵) ≠ ∅)
2	1	adantr 480	. . 3 ⊢ ((𝐶 ∈ (𝐴(,)𝐵) ∧ 𝐷 ∈ (𝐴(,)𝐵)) → (𝐴(,)𝐵) ≠ ∅)
3		ndmioo 13436	. . . 4 ⊢ (¬ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴(,)𝐵) = ∅)
4	3	necon1ai 2974	. . 3 ⊢ ((𝐴(,)𝐵) ≠ ∅ → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*))
5	2, 4	syl 17	. 2 ⊢ ((𝐶 ∈ (𝐴(,)𝐵) ∧ 𝐷 ∈ (𝐴(,)𝐵)) → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*))
6		eliooord 13468	. . . 4 ⊢ (𝐶 ∈ (𝐴(,)𝐵) → (𝐴 < 𝐶 ∧ 𝐶 < 𝐵))
7	6	adantr 480	. . 3 ⊢ ((𝐶 ∈ (𝐴(,)𝐵) ∧ 𝐷 ∈ (𝐴(,)𝐵)) → (𝐴 < 𝐶 ∧ 𝐶 < 𝐵))
8	7	simpld 494	. 2 ⊢ ((𝐶 ∈ (𝐴(,)𝐵) ∧ 𝐷 ∈ (𝐴(,)𝐵)) → 𝐴 < 𝐶)
9		eliooord 13468	. . . 4 ⊢ (𝐷 ∈ (𝐴(,)𝐵) → (𝐴 < 𝐷 ∧ 𝐷 < 𝐵))
10	9	adantl 481	. . 3 ⊢ ((𝐶 ∈ (𝐴(,)𝐵) ∧ 𝐷 ∈ (𝐴(,)𝐵)) → (𝐴 < 𝐷 ∧ 𝐷 < 𝐵))
11	10	simprd 495	. 2 ⊢ ((𝐶 ∈ (𝐴(,)𝐵) ∧ 𝐷 ∈ (𝐴(,)𝐵)) → 𝐷 < 𝐵)
12		iccssioo 13478	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐶 ∧ 𝐷 < 𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴(,)𝐵))
13	5, 8, 11, 12	syl12anc 836	1 ⊢ ((𝐶 ∈ (𝐴(,)𝐵) ∧ 𝐷 ∈ (𝐴(,)𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴(,)𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 ≠ wne 2946 ⊆ wss 3976 ∅c0 4352 class class class wbr 5166 (class class class)co 7450 ℝ^*cxr 11325 < clt 11326 (,)cioo 13409 [,]cicc 13412

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 ax-cnex 11242 ax-resscn 11243 ax-pre-lttri 11260 ax-pre-lttrn 11261

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-po 5607 df-so 5608 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-ov 7453 df-oprab 7454 df-mpo 7455 df-1st 8032 df-2nd 8033 df-er 8765 df-en 9006 df-dom 9007 df-sdom 9008 df-pnf 11328 df-mnf 11329 df-xr 11330 df-ltxr 11331 df-le 11332 df-ioo 13413 df-icc 13416

This theorem is referenced by: dvivthlem1 26069 dvivthlem2 26070 amgmlem 27053 ioosconn 35217 aks4d1p1p5 42034 amgmwlem 48902

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