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Theorem ico0 13054

Description: An empty open interval of extended reals. (Contributed by FL, 30-May-2014.)

**Assertion**
Ref	Expression
ico0	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴[,)𝐵) = ∅ ↔ 𝐵 ≤ 𝐴))

Proof of Theorem ico0 Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		icoval 13046	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴[,)𝐵) = {𝑥 ∈ ℝ^ ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)})
2	1	eqeq1d 2740	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ((𝐴[,)𝐵) = ∅ ↔ {𝑥 ∈ ℝ^ ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)} = ∅))
3		df-ne 2943	. . . . . 6 ⊢ ({𝑥 ∈ ℝ^* ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)} ≠ ∅ ↔ ¬ {𝑥 ∈ ℝ^* ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)} = ∅)
4		rabn0 4316	. . . . . 6 ⊢ ({𝑥 ∈ ℝ^* ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)} ≠ ∅ ↔ ∃𝑥 ∈ ℝ^* (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵))
5	3, 4	bitr3i 276	. . . . 5 ⊢ (¬ {𝑥 ∈ ℝ^* ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)} = ∅ ↔ ∃𝑥 ∈ ℝ^* (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵))
6		xrlelttr 12819	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝑥 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵) → 𝐴 < 𝐵))
7	6	3com23 1124	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑥 ∈ ℝ^*) → ((𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵) → 𝐴 < 𝐵))
8	7	3expa 1116	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑥 ∈ ℝ^) → ((𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵) → 𝐴 < 𝐵))
9	8	rexlimdva 3212	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (∃𝑥 ∈ ℝ^ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵) → 𝐴 < 𝐵))
10		qbtwnxr 12863	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵))
11		qre 12622	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℚ → 𝑥 ∈ ℝ)
12	11	rexrd 10956	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℚ → 𝑥 ∈ ℝ^*)
13	12	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ^* ∧ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵)) → (𝑥 ∈ ℚ → 𝑥 ∈ ℝ^*))
14		simpr1 1192	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ^* ∧ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵)) → 𝐴 ∈ ℝ^*)
15		simpl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ^* ∧ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵)) → 𝑥 ∈ ℝ^*)
16		xrltle 12812	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝑥 ∈ ℝ^*) → (𝐴 < 𝑥 → 𝐴 ≤ 𝑥))
17	14, 15, 16	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ^* ∧ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵)) → (𝐴 < 𝑥 → 𝐴 ≤ 𝑥))
18	17	anim1d 610	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ^* ∧ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵)) → ((𝐴 < 𝑥 ∧ 𝑥 < 𝐵) → (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)))
19	13, 18	anim12d 608	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ^* ∧ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵)) → ((𝑥 ∈ ℚ ∧ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)) → (𝑥 ∈ ℝ^* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵))))
20	19	ex 412	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ^* → ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → ((𝑥 ∈ ℚ ∧ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)) → (𝑥 ∈ ℝ^* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)))))
21	12, 20	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℚ → ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → ((𝑥 ∈ ℚ ∧ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)) → (𝑥 ∈ ℝ^* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)))))
22	21	adantr 480	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℚ ∧ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)) → ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → ((𝑥 ∈ ℚ ∧ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)) → (𝑥 ∈ ℝ^* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)))))
23	22	pm2.43b 55	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → ((𝑥 ∈ ℚ ∧ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)) → (𝑥 ∈ ℝ^* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵))))
24	23	reximdv2 3198	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → (∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵) → ∃𝑥 ∈ ℝ^* (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)))
25	10, 24	mpd 15	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → ∃𝑥 ∈ ℝ^* (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵))
26	25	3expia 1119	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℝ^ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)))
27	9, 26	impbid 211	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (∃𝑥 ∈ ℝ^ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵) ↔ 𝐴 < 𝐵))
28	5, 27	syl5bb 282	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (¬ {𝑥 ∈ ℝ^ ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)} = ∅ ↔ 𝐴 < 𝐵))
29		xrltnle 10973	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴))
30	28, 29	bitrd 278	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (¬ {𝑥 ∈ ℝ^ ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)} = ∅ ↔ ¬ 𝐵 ≤ 𝐴))
31	30	con4bid 316	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ({𝑥 ∈ ℝ^ ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)} = ∅ ↔ 𝐵 ≤ 𝐴))
32	2, 31	bitrd 278	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴[,)𝐵) = ∅ ↔ 𝐵 ≤ 𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∃wrex 3064 {crab 3067 ∅c0 4253 class class class wbr 5070 (class class class)co 7255 ℝ^*cxr 10939 < clt 10940 ≤ cle 10941 ℚcq 12617 [,)cico 13010

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-sup 9131 df-inf 9132 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-n0 12164 df-z 12250 df-uz 12512 df-q 12618 df-ico 13014

This theorem is referenced by: icombl 24633 ioombl 24634 difioo 31005 volico 43414 voliooico 43423 voliccico 43430 ovn0lem 43993 ovnhoilem1 44029 hspmbllem1 44054

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