elioc2 - Metamath Proof Explorer

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Theorem elioc2 13406

Description: Membership in an open-below, closed-above real interval. (Contributed by Paul Chapman, 30-Dec-2007.) (Revised by Mario Carneiro, 14-Jun-2014.)

**Assertion**
Ref	Expression
elioc2	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)))

**Proof of Theorem elioc2**
Step	Hyp	Ref	Expression
1		rexr 11221	. . 3 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ^*)
2		elioc1 13384	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ^ ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)))
3	1, 2	sylan2 602	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ^* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)))
4		mnfxr 11232	. . . . . . . 8 ⊢ -∞ ∈ ℝ^*
5	4	a1i 11	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)) → -∞ ∈ ℝ^*)
6		simpll 776	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)) → 𝐴 ∈ ℝ^*)
7		simpr1 1207	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)) → 𝐶 ∈ ℝ^*)
8		mnfle 13130	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ^* → -∞ ≤ 𝐴)
9	8	ad2antrr 736	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)) → -∞ ≤ 𝐴)
10		simpr2 1208	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)) → 𝐴 < 𝐶)
11	5, 6, 7, 9, 10	xrlelttrd 13155	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)) → -∞ < 𝐶)
12	1	ad2antlr 737	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)) → 𝐵 ∈ ℝ^*)
13		pnfxr 11229	. . . . . . . 8 ⊢ +∞ ∈ ℝ^*
14	13	a1i 11	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)) → +∞ ∈ ℝ^*)
15		simpr3 1209	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)) → 𝐶 ≤ 𝐵)
16		ltpnf 13115	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ → 𝐵 < +∞)
17	16	ad2antlr 737	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)) → 𝐵 < +∞)
18	7, 12, 14, 15, 17	xrlelttrd 13155	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)) → 𝐶 < +∞)
19		xrrebnd 13164	. . . . . . 7 ⊢ (𝐶 ∈ ℝ^* → (𝐶 ∈ ℝ ↔ (-∞ < 𝐶 ∧ 𝐶 < +∞)))
20	7, 19	syl 17	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)) → (𝐶 ∈ ℝ ↔ (-∞ < 𝐶 ∧ 𝐶 < +∞)))
21	11, 18, 20	mpbir2and 723	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)) → 𝐶 ∈ ℝ)
22	21, 10, 15	3jca 1140	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)) → (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵))
23	22	ex 416	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → ((𝐶 ∈ ℝ^* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵) → (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)))
24		rexr 11221	. . . 4 ⊢ (𝐶 ∈ ℝ → 𝐶 ∈ ℝ^*)
25	24	3anim1i 1164	. . 3 ⊢ ((𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵) → (𝐶 ∈ ℝ^* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵))
26	23, 25	impbid1 227	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → ((𝐶 ∈ ℝ^* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)))
27	3, 26	bitrd 281	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 399 ∧ w3a 1097 ∈ wcel 2141 class class class wbr 5097 (class class class)co 7390 ℝcr 11065 +∞cpnf 11206 -∞cmnf 11207 ℝ^*cxr 11208 < clt 11209 ≤ cle 11210 (,]cioc 13343

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7712 ax-cnex 11122 ax-resscn 11123 ax-pre-lttri 11140 ax-pre-lttrn 11141

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-br 5098 df-opab 5160 df-mpt 5179 df-id 5538 df-po 5551 df-so 5552 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-iota 6471 df-fun 6517 df-fn 6518 df-f 6519 df-f1 6520 df-fo 6521 df-f1o 6522 df-fv 6523 df-ov 7393 df-oprab 7394 df-mpo 7395 df-er 8671 df-en 8921 df-dom 8922 df-sdom 8923 df-pnf 11211 df-mnf 11212 df-xr 11213 df-ltxr 11214 df-le 11215 df-ioc 13347

This theorem is referenced by: iocssre 13424 ef01bndlem 16206 sin01bnd 16207 cos01bnd 16208 cos1bnd 16209 sinltx 16211 sin01gt0 16212 cos01gt0 16213 sin02gt0 16214 sincos1sgn 16215 sincos2sgn 16216 icoopnst 24988 iocopnst 24989 ismbf3d 25703 aaliou3lem2 26394 aaliou3lem3 26395 pilem2 26502 sinhalfpilem 26515 sincosq1lem 26549 coseq0negpitopi 26555 tangtx 26557 sincos4thpi 26565 efif1olem1 26594 efif1olem2 26595 efif1o 26598 efifo 26599 ellogrn 26611 logimclad 26624 ellogdm 26691 logdmnrp 26693 dvloglem 26700 dvlog2lem 26704 asinneg 26938 atans2 26983 ressatans 26986 abvcxp 27666 ostth2 27688 xrge0iifcv 34191 xrge0iifiso 34192 xrge0iifhom 34194 sinccvglem 35982 bj-pinftyccb 37673 bj-pinftynminfty 37679 dvasin 38163 areacirclem4 38170 gtnelioc 46027 limcicciooub 46171 fourierdlem4 46645 fourierdlem26 46667 fourierdlem33 46674 fourierdlem37 46678 fourierdlem65 46705 fourierdlem79 46719 fouriersw 46765 eenglngeehlnmlem1 49319 eenglngeehlnmlem2 49320 io1ii 49502 sepfsepc 49509

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