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Theorem ico0 13515

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 13507	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴[,)𝐵) = {𝑥 ∈ ℝ^ ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)})
2	1	eqeq1d 2763	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ((𝐴[,)𝐵) = ∅ ↔ {𝑥 ∈ ℝ^ ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)} = ∅))
3		df-ne 2957	. . . . . 6 ⊢ ({𝑥 ∈ ℝ^* ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)} ≠ ∅ ↔ ¬ {𝑥 ∈ ℝ^* ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)} = ∅)
4		rabn0 4339	. . . . . 6 ⊢ ({𝑥 ∈ ℝ^* ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)} ≠ ∅ ↔ ∃𝑥 ∈ ℝ^* (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵))
5	3, 4	bitr3i 280	. . . . 5 ⊢ (¬ {𝑥 ∈ ℝ^* ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)} = ∅ ↔ ∃𝑥 ∈ ℝ^* (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵))
6		xrlelttr 13278	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝑥 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵) → 𝐴 < 𝐵))
7	6	3com23 1144	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑥 ∈ ℝ^*) → ((𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵) → 𝐴 < 𝐵))
8	7	3expa 1136	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑥 ∈ ℝ^) → ((𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵) → 𝐴 < 𝐵))
9	8	rexlimdva 3164	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (∃𝑥 ∈ ℝ^ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵) → 𝐴 < 𝐵))
10		qbtwnxr 13323	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵))
11		qre 13073	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℚ → 𝑥 ∈ ℝ)
12	11	rexrd 11352	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℚ → 𝑥 ∈ ℝ^*)
13	12	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ^* ∧ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵)) → (𝑥 ∈ ℚ → 𝑥 ∈ ℝ^*))
14		simpr1 1213	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ^* ∧ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵)) → 𝐴 ∈ ℝ^*)
15		simpl 488	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ^* ∧ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵)) → 𝑥 ∈ ℝ^*)
16		xrltle 13271	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝑥 ∈ ℝ^*) → (𝐴 < 𝑥 → 𝐴 ≤ 𝑥))
17	14, 15, 16	syl2anc 596	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ^* ∧ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵)) → (𝐴 < 𝑥 → 𝐴 ≤ 𝑥))
18	17	anim1d 623	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ^* ∧ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵)) → ((𝐴 < 𝑥 ∧ 𝑥 < 𝐵) → (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)))
19	13, 18	anim12d 621	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ^* ∧ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵)) → ((𝑥 ∈ ℚ ∧ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)) → (𝑥 ∈ ℝ^* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵))))
20	19	ex 418	. . . . . . . . . . . 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 18	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℚ → ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → ((𝑥 ∈ ℚ ∧ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)) → (𝑥 ∈ ℝ^* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)))))
22	21	adantr 486	. . . . . . . . . 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 56	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → ((𝑥 ∈ ℚ ∧ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)) → (𝑥 ∈ ℝ^* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵))))
24	23	reximdv2 3173	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → (∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵) → ∃𝑥 ∈ ℝ^* (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)))
25	10, 24	mpd 16	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → ∃𝑥 ∈ ℝ^* (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵))
26	25	3expia 1139	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℝ^ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)))
27	9, 26	impbid 215	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (∃𝑥 ∈ ℝ^ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵) ↔ 𝐴 < 𝐵))
28	5, 27	bitrid 286	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (¬ {𝑥 ∈ ℝ^ ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)} = ∅ ↔ 𝐴 < 𝐵))
29		xrltnle 11369	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴))
30	28, 29	bitrd 282	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (¬ {𝑥 ∈ ℝ^ ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)} = ∅ ↔ ¬ 𝐵 ≤ 𝐴))
31	30	con4bid 320	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ({𝑥 ∈ ℝ^ ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)} = ∅ ↔ 𝐵 ≤ 𝐴))
32	2, 31	bitrd 282	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴[,)𝐵) = ∅ ↔ 𝐵 ≤ 𝐴))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2956 ∃wrex 3087 {crab 3413 ∅c0 4279 class class class wbr 5103 (class class class)co 7418 ℝ^*cxr 11335 < clt 11336 ≤ cle 11337 ℚcq 13068 [,)cico 13471

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7749 ax-cnex 11249 ax-resscn 11250 ax-1cn 11251 ax-icn 11252 ax-addcl 11253 ax-addrcl 11254 ax-mulcl 11255 ax-mulrcl 11256 ax-mulcom 11257 ax-addass 11258 ax-mulass 11259 ax-distr 11260 ax-i2m1 11261 ax-1ne0 11262 ax-1rid 11263 ax-rnegex 11264 ax-rrecex 11265 ax-cnre 11266 ax-pre-lttri 11267 ax-pre-lttrn 11268 ax-pre-ltadd 11269 ax-pre-mulgt0 11270 ax-pre-sup 11271

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7375 df-ov 7421 df-oprab 7422 df-mpo 7423 df-om 7876 df-1st 7999 df-2nd 8000 df-frecs 8292 df-wrecs 8323 df-recs 8372 df-rdg 8411 df-er 8710 df-en 8967 df-dom 8968 df-sdom 8969 df-sup 9427 df-inf 9428 df-pnf 11338 df-mnf 11339 df-xr 11340 df-ltxr 11341 df-le 11342 df-sub 11536 df-neg 11537 df-div 11967 df-nn 12329 df-n0 12600 df-z 12687 df-uz 12959 df-q 13069 df-ico 13475

This theorem is used by: icombl 25878 ioombl 25879 difioo 33367 volico 46962 voliooico 46971 voliccico 46978 ovn0lem 47544 ovnhoilem1 47580 hspmbllem1 47605

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