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Theorem ico0 13286

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 13278	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴[,)𝐵) = {𝑥 ∈ ℝ^ ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)})
2	1	eqeq1d 2733	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ((𝐴[,)𝐵) = ∅ ↔ {𝑥 ∈ ℝ^ ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)} = ∅))
3		df-ne 2929	. . . . . 6 ⊢ ({𝑥 ∈ ℝ^* ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)} ≠ ∅ ↔ ¬ {𝑥 ∈ ℝ^* ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)} = ∅)
4		rabn0 4334	. . . . . 6 ⊢ ({𝑥 ∈ ℝ^* ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)} ≠ ∅ ↔ ∃𝑥 ∈ ℝ^* (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵))
5	3, 4	bitr3i 277	. . . . 5 ⊢ (¬ {𝑥 ∈ ℝ^* ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)} = ∅ ↔ ∃𝑥 ∈ ℝ^* (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵))
6		xrlelttr 13050	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝑥 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵) → 𝐴 < 𝐵))
7	6	3com23 1126	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑥 ∈ ℝ^*) → ((𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵) → 𝐴 < 𝐵))
8	7	3expa 1118	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑥 ∈ ℝ^) → ((𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵) → 𝐴 < 𝐵))
9	8	rexlimdva 3133	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (∃𝑥 ∈ ℝ^ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵) → 𝐴 < 𝐵))
10		qbtwnxr 13094	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵))
11		qre 12846	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℚ → 𝑥 ∈ ℝ)
12	11	rexrd 11157	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℚ → 𝑥 ∈ ℝ^*)
13	12	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ^* ∧ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵)) → (𝑥 ∈ ℚ → 𝑥 ∈ ℝ^*))
14		simpr1 1195	. . . . . . . . . . . . . . . 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 13043	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝑥 ∈ ℝ^*) → (𝐴 < 𝑥 → 𝐴 ≤ 𝑥))
17	14, 15, 16	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ^* ∧ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵)) → (𝐴 < 𝑥 → 𝐴 ≤ 𝑥))
18	17	anim1d 611	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ^* ∧ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵)) → ((𝐴 < 𝑥 ∧ 𝑥 < 𝐵) → (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)))
19	13, 18	anim12d 609	. . . . . . . . . . . . 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 3142	. . . . . . . 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 1121	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℝ^ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)))
27	9, 26	impbid 212	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (∃𝑥 ∈ ℝ^ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵) ↔ 𝐴 < 𝐵))
28	5, 27	bitrid 283	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (¬ {𝑥 ∈ ℝ^ ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)} = ∅ ↔ 𝐴 < 𝐵))
29		xrltnle 11174	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴))
30	28, 29	bitrd 279	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (¬ {𝑥 ∈ ℝ^ ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)} = ∅ ↔ ¬ 𝐵 ≤ 𝐴))
31	30	con4bid 317	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ({𝑥 ∈ ℝ^ ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)} = ∅ ↔ 𝐵 ≤ 𝐴))
32	2, 31	bitrd 279	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴[,)𝐵) = ∅ ↔ 𝐵 ≤ 𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 ∃wrex 3056 {crab 3395 ∅c0 4278 class class class wbr 5086 (class class class)co 7341 ℝ^*cxr 11140 < clt 11141 ≤ cle 11142 ℚcq 12841 [,)cico 13242

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5229 ax-nul 5239 ax-pow 5298 ax-pr 5365 ax-un 7663 ax-cnex 11057 ax-resscn 11058 ax-1cn 11059 ax-icn 11060 ax-addcl 11061 ax-addrcl 11062 ax-mulcl 11063 ax-mulrcl 11064 ax-mulcom 11065 ax-addass 11066 ax-mulass 11067 ax-distr 11068 ax-i2m1 11069 ax-1ne0 11070 ax-1rid 11071 ax-rnegex 11072 ax-rrecex 11073 ax-cnre 11074 ax-pre-lttri 11075 ax-pre-lttrn 11076 ax-pre-ltadd 11077 ax-pre-mulgt0 11078 ax-pre-sup 11079

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4279 df-if 4471 df-pw 4547 df-sn 4572 df-pr 4574 df-op 4578 df-uni 4855 df-iun 4938 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5506 df-eprel 5511 df-po 5519 df-so 5520 df-fr 5564 df-we 5566 df-xp 5617 df-rel 5618 df-cnv 5619 df-co 5620 df-dm 5621 df-rn 5622 df-res 5623 df-ima 5624 df-pred 6243 df-ord 6304 df-on 6305 df-lim 6306 df-suc 6307 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-om 7792 df-1st 7916 df-2nd 7917 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-er 8617 df-en 8865 df-dom 8866 df-sdom 8867 df-sup 9321 df-inf 9322 df-pnf 11143 df-mnf 11144 df-xr 11145 df-ltxr 11146 df-le 11147 df-sub 11341 df-neg 11342 df-div 11770 df-nn 12121 df-n0 12377 df-z 12464 df-uz 12728 df-q 12842 df-ico 13246

This theorem is referenced by: icombl 25487 ioombl 25488 difioo 32757 volico 46021 voliooico 46030 voliccico 46037 ovn0lem 46603 ovnhoilem1 46639 hspmbllem1 46664

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