iccssioo2 - Metamath Proof Explorer

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Theorem iccssioo2 12973

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 4235	. . . 4 ⊢ (𝐶 ∈ (𝐴(,)𝐵) → (𝐴(,)𝐵) ≠ ∅)
2	1	adantr 484	. . 3 ⊢ ((𝐶 ∈ (𝐴(,)𝐵) ∧ 𝐷 ∈ (𝐴(,)𝐵)) → (𝐴(,)𝐵) ≠ ∅)
3		ndmioo 12927	. . . 4 ⊢ (¬ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴(,)𝐵) = ∅)
4	3	necon1ai 2959	. . 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 12959	. . . 4 ⊢ (𝐶 ∈ (𝐴(,)𝐵) → (𝐴 < 𝐶 ∧ 𝐶 < 𝐵))
7	6	adantr 484	. . 3 ⊢ ((𝐶 ∈ (𝐴(,)𝐵) ∧ 𝐷 ∈ (𝐴(,)𝐵)) → (𝐴 < 𝐶 ∧ 𝐶 < 𝐵))
8	7	simpld 498	. 2 ⊢ ((𝐶 ∈ (𝐴(,)𝐵) ∧ 𝐷 ∈ (𝐴(,)𝐵)) → 𝐴 < 𝐶)
9		eliooord 12959	. . . 4 ⊢ (𝐷 ∈ (𝐴(,)𝐵) → (𝐴 < 𝐷 ∧ 𝐷 < 𝐵))
10	9	adantl 485	. . 3 ⊢ ((𝐶 ∈ (𝐴(,)𝐵) ∧ 𝐷 ∈ (𝐴(,)𝐵)) → (𝐴 < 𝐷 ∧ 𝐷 < 𝐵))
11	10	simprd 499	. 2 ⊢ ((𝐶 ∈ (𝐴(,)𝐵) ∧ 𝐷 ∈ (𝐴(,)𝐵)) → 𝐷 < 𝐵)
12		iccssioo 12969	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐶 ∧ 𝐷 < 𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴(,)𝐵))
13	5, 8, 11, 12	syl12anc 837	1 ⊢ ((𝐶 ∈ (𝐴(,)𝐵) ∧ 𝐷 ∈ (𝐴(,)𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴(,)𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2112 ≠ wne 2932 ⊆ wss 3853 ∅c0 4223 class class class wbr 5039 (class class class)co 7191 ℝ^*cxr 10831 < clt 10832 (,)cioo 12900 [,]cicc 12903

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-pre-lttri 10768 ax-pre-lttrn 10769

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-po 5453 df-so 5454 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-1st 7739 df-2nd 7740 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-ioo 12904 df-icc 12907

This theorem is referenced by: dvivthlem1 24859 dvivthlem2 24860 amgmlem 25826 ioosconn 32876 aks4d1p1p5 39765 amgmwlem 46120

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