iooss1 - Metamath Proof Explorer

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Theorem iooss1 13392

Description: Subset relationship for open intervals of extended reals. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 20-Feb-2015.)

**Assertion**
Ref	Expression
iooss1	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → (𝐵(,)𝐶) ⊆ (𝐴(,)𝐶))

Proof of Theorem iooss1 Dummy variables 𝑥 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ioo 13361	. 2 ⊢ (,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)})
2		xrlelttr 13168	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → ((𝐴 ≤ 𝐵 ∧ 𝐵 < 𝑤) → 𝐴 < 𝑤))
3	1, 1, 2	ixxss1 13375	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → (𝐵(,)𝐶) ⊆ (𝐴(,)𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2099 ⊆ wss 3947 class class class wbr 5148 (class class class)co 7420 ℝ^*cxr 11278 < clt 11279 ≤ cle 11280 (,)cioo 13357

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-pre-lttri 11213 ax-pre-lttrn 11214

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 7993 df-2nd 7994 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-ioo 13361

This theorem is referenced by: ioodisj 13492 tgqioo 24729 ioorcl2 25514 itg2gt0 25703 itgsplitioo 25780 ditgcl 25800 ditgswap 25801 ditgsplitlem 25802 dvferm1lem 25929 dvferm 25933 dvlip 25939 dvgt0lem1 25948 dvivthlem1 25954 lhop1lem 25959 lhop2 25961 dvcvx 25966 dvfsumle 25967 dvfsumleOLD 25968 dvfsumge 25969 dvfsumabs 25970 ftc1lem1 25983 ftc1a 25985 ftc1lem4 25987 ftc2ditglem 25993 tanregt0 26486 basellem4 27029 pntlemp 27556 radcnvrat 43751 limcresiooub 45030 fourierdlem46 45540 fourierdlem48 45542 fourierdlem49 45543 fourierdlem74 45568 fourierdlem104 45598 fourierdlem113 45607 fouriersw 45619

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