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Theorem ico0 13352

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 13344	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴[,)𝐵) = {𝑥 ∈ ℝ^ ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)})
2	1	eqeq1d 2731	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ((𝐴[,)𝐵) = ∅ ↔ {𝑥 ∈ ℝ^ ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)} = ∅))
3		df-ne 2926	. . . . . 6 ⊢ ({𝑥 ∈ ℝ^* ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)} ≠ ∅ ↔ ¬ {𝑥 ∈ ℝ^* ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)} = ∅)
4		rabn0 4352	. . . . . 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 13116	. . . . . . . . 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 3134	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (∃𝑥 ∈ ℝ^ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵) → 𝐴 < 𝐵))
10		qbtwnxr 13160	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵))
11		qre 12912	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℚ → 𝑥 ∈ ℝ)
12	11	rexrd 11224	. . . . . . . . . . . 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 13109	. . . . . . . . . . . . . . . 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 3143	. . . . . . . 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 11241	. . . 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 1540 ∈ wcel 2109 ≠ wne 2925 ∃wrex 3053 {crab 3405 ∅c0 4296 class class class wbr 5107 (class class class)co 7387 ℝ^*cxr 11207 < clt 11208 ≤ cle 11209 ℚcq 12907 [,)cico 13308

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-er 8671 df-en 8919 df-dom 8920 df-sdom 8921 df-sup 9393 df-inf 9394 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-n0 12443 df-z 12530 df-uz 12794 df-q 12908 df-ico 13312

This theorem is referenced by: icombl 25465 ioombl 25466 difioo 32705 volico 45981 voliooico 45990 voliccico 45997 ovn0lem 46563 ovnhoilem1 46599 hspmbllem1 46624

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