iccssioo2 - Metamath Proof Explorer

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Theorem iccssioo2 13245

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 4280	. . . 4 ⊢ (𝐶 ∈ (𝐴(,)𝐵) → (𝐴(,)𝐵) ≠ ∅)
2	1	adantr 481	. . 3 ⊢ ((𝐶 ∈ (𝐴(,)𝐵) ∧ 𝐷 ∈ (𝐴(,)𝐵)) → (𝐴(,)𝐵) ≠ ∅)
3		ndmioo 13199	. . . 4 ⊢ (¬ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴(,)𝐵) = ∅)
4	3	necon1ai 2968	. . 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 13231	. . . 4 ⊢ (𝐶 ∈ (𝐴(,)𝐵) → (𝐴 < 𝐶 ∧ 𝐶 < 𝐵))
7	6	adantr 481	. . 3 ⊢ ((𝐶 ∈ (𝐴(,)𝐵) ∧ 𝐷 ∈ (𝐴(,)𝐵)) → (𝐴 < 𝐶 ∧ 𝐶 < 𝐵))
8	7	simpld 495	. 2 ⊢ ((𝐶 ∈ (𝐴(,)𝐵) ∧ 𝐷 ∈ (𝐴(,)𝐵)) → 𝐴 < 𝐶)
9		eliooord 13231	. . . 4 ⊢ (𝐷 ∈ (𝐴(,)𝐵) → (𝐴 < 𝐷 ∧ 𝐷 < 𝐵))
10	9	adantl 482	. . 3 ⊢ ((𝐶 ∈ (𝐴(,)𝐵) ∧ 𝐷 ∈ (𝐴(,)𝐵)) → (𝐴 < 𝐷 ∧ 𝐷 < 𝐵))
11	10	simprd 496	. 2 ⊢ ((𝐶 ∈ (𝐴(,)𝐵) ∧ 𝐷 ∈ (𝐴(,)𝐵)) → 𝐷 < 𝐵)
12		iccssioo 13241	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐶 ∧ 𝐷 < 𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴(,)𝐵))
13	5, 8, 11, 12	syl12anc 834	1 ⊢ ((𝐶 ∈ (𝐴(,)𝐵) ∧ 𝐷 ∈ (𝐴(,)𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴(,)𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2105 ≠ wne 2940 ⊆ wss 3897 ∅c0 4268 class class class wbr 5089 (class class class)co 7329 ℝ^*cxr 11101 < clt 11102 (,)cioo 13172 [,]cicc 13175

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5240 ax-nul 5247 ax-pow 5305 ax-pr 5369 ax-un 7642 ax-cnex 11020 ax-resscn 11021 ax-pre-lttri 11038 ax-pre-lttrn 11039

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3404 df-v 3443 df-sbc 3727 df-csb 3843 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4269 df-if 4473 df-pw 4548 df-sn 4573 df-pr 4575 df-op 4579 df-uni 4852 df-iun 4940 df-br 5090 df-opab 5152 df-mpt 5173 df-id 5512 df-po 5526 df-so 5527 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-iota 6425 df-fun 6475 df-fn 6476 df-f 6477 df-f1 6478 df-fo 6479 df-f1o 6480 df-fv 6481 df-ov 7332 df-oprab 7333 df-mpo 7334 df-1st 7891 df-2nd 7892 df-er 8561 df-en 8797 df-dom 8798 df-sdom 8799 df-pnf 11104 df-mnf 11105 df-xr 11106 df-ltxr 11107 df-le 11108 df-ioo 13176 df-icc 13179

This theorem is referenced by: dvivthlem1 25270 dvivthlem2 25271 amgmlem 26237 ioosconn 33449 aks4d1p1p5 40330 amgmwlem 46846

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