iccssioo2 - Intuitionistic Logic Explorer

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Theorem iccssioo2 10180

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		eliooxr 10161	. . 3 ⊢ (𝐶 ∈ (𝐴(,)𝐵) → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*))
2	1	adantr 276	. 2 ⊢ ((𝐶 ∈ (𝐴(,)𝐵) ∧ 𝐷 ∈ (𝐴(,)𝐵)) → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*))
3		eliooord 10162	. . . 4 ⊢ (𝐶 ∈ (𝐴(,)𝐵) → (𝐴 < 𝐶 ∧ 𝐶 < 𝐵))
4	3	adantr 276	. . 3 ⊢ ((𝐶 ∈ (𝐴(,)𝐵) ∧ 𝐷 ∈ (𝐴(,)𝐵)) → (𝐴 < 𝐶 ∧ 𝐶 < 𝐵))
5	4	simpld 112	. 2 ⊢ ((𝐶 ∈ (𝐴(,)𝐵) ∧ 𝐷 ∈ (𝐴(,)𝐵)) → 𝐴 < 𝐶)
6		eliooord 10162	. . . 4 ⊢ (𝐷 ∈ (𝐴(,)𝐵) → (𝐴 < 𝐷 ∧ 𝐷 < 𝐵))
7	6	adantl 277	. . 3 ⊢ ((𝐶 ∈ (𝐴(,)𝐵) ∧ 𝐷 ∈ (𝐴(,)𝐵)) → (𝐴 < 𝐷 ∧ 𝐷 < 𝐵))
8	7	simprd 114	. 2 ⊢ ((𝐶 ∈ (𝐴(,)𝐵) ∧ 𝐷 ∈ (𝐴(,)𝐵)) → 𝐷 < 𝐵)
9		iccssioo 10176	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐶 ∧ 𝐷 < 𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴(,)𝐵))
10	2, 5, 8, 9	syl12anc 1271	1 ⊢ ((𝐶 ∈ (𝐴(,)𝐵) ∧ 𝐷 ∈ (𝐴(,)𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴(,)𝐵))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ∈ wcel 2202 ⊆ wss 3200 class class class wbr 4088 (class class class)co 6017 ℝ^*cxr 8212 < clt 8213 (,)cioo 10122 [,]cicc 10125

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-sep 4207 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-cnex 8122 ax-resscn 8123 ax-pre-ltirr 8143 ax-pre-ltwlin 8144 ax-pre-lttrn 8145

This theorem depends on definitions: df-bi 117 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-rab 2519 df-v 2804 df-sbc 3032 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-br 4089 df-opab 4151 df-id 4390 df-po 4393 df-iso 4394 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-iota 5286 df-fun 5328 df-fv 5334 df-ov 6020 df-oprab 6021 df-mpo 6022 df-pnf 8215 df-mnf 8216 df-xr 8217 df-ltxr 8218 df-le 8219 df-ioo 10126 df-icc 10129

This theorem is referenced by: (None)

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