icossre - Metamath Proof Explorer

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Theorem icossre 13437

Description: A closed-below interval with real lower bound is a set of reals. (Contributed by Mario Carneiro, 14-Jun-2014.)

**Assertion**
Ref	Expression
icossre	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → (𝐴[,)𝐵) ⊆ ℝ)

Proof of Theorem icossre Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elico2 13420	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → (𝑥 ∈ (𝐴[,)𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)))
2	1	biimp3a 1465	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝑥 ∈ (𝐴[,)𝐵)) → (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵))
3	2	simp1d 1139	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^* ∧ 𝑥 ∈ (𝐴[,)𝐵)) → 𝑥 ∈ ℝ)
4	3	3expia 1118	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → (𝑥 ∈ (𝐴[,)𝐵) → 𝑥 ∈ ℝ))
5	4	ssrdv 3983	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → (𝐴[,)𝐵) ⊆ ℝ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 ∈ wcel 2098 ⊆ wss 3945 class class class wbr 5148 (class class class)co 7417 ℝcr 11137 ℝ^*cxr 11277 < clt 11278 ≤ cle 11279 [,)cico 13358

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5299 ax-nul 5306 ax-pow 5364 ax-pr 5428 ax-un 7739 ax-cnex 11194 ax-resscn 11195 ax-pre-lttri 11212 ax-pre-lttrn 11213

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5575 df-po 5589 df-so 5590 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6499 df-fun 6549 df-fn 6550 df-f 6551 df-f1 6552 df-fo 6553 df-f1o 6554 df-fv 6555 df-ov 7420 df-oprab 7421 df-mpo 7422 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-ico 13362

This theorem is referenced by: icoshftf1o 13483 ico01fl0 13817 rexico 15333 rlim3 15475 fprodge1 15972 ovolicopnf 25490 dvfsumrlim2 26004 tanord1 26508 chebbnd1 27442 chebbnd2 27447 dchrisumlem3 27461 pntpbnd1 27556 pntibndlem2 27561 sxbrsigalem0 33978 dya2iocress 33981 dya2iocucvr 33991 sitmcl 34058 tan2h 37172 icoopn 44990 limciccioolb 45089 ltmod 45106 limcresioolb 45111 limsupresre 45164 limsupresico 45168 liminfresico 45239 fourierdlem32 45607 fourierdlem46 45620 fourierdlem48 45622 fourierdlem93 45667 fouriersw 45699 fouriercn 45700 hoissre 46012 hoissrrn2 46046 hoidmv1lelem2 46060 ovnlecvr2 46078 hspdifhsp 46084 hoiqssbllem2 46091 hspmbllem2 46095 iinhoiicclem 46141

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