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Theorem ico0 13389

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 13381	. . 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 4340	. . . . . 6 ⊢ ({𝑥 ∈ ℝ^* ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)} ≠ ∅ ↔ ∃𝑥 ∈ ℝ^* (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵))
5	3, 4	bitr3i 279	. . . . 5 ⊢ (¬ {𝑥 ∈ ℝ^* ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)} = ∅ ↔ ∃𝑥 ∈ ℝ^* (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵))
6		xrlelttr 13152	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝑥 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵) → 𝐴 < 𝐵))
7	6	3com23 1138	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑥 ∈ ℝ^*) → ((𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵) → 𝐴 < 𝐵))
8	7	3expa 1130	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑥 ∈ ℝ^) → ((𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵) → 𝐴 < 𝐵))
9	8	rexlimdva 3162	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (∃𝑥 ∈ ℝ^ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵) → 𝐴 < 𝐵))
10		qbtwnxr 13197	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵))
11		qre 12948	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℚ → 𝑥 ∈ ℝ)
12	11	rexrd 11226	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℚ → 𝑥 ∈ ℝ^*)
13	12	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ^* ∧ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵)) → (𝑥 ∈ ℚ → 𝑥 ∈ ℝ^*))
14		simpr1 1207	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ^* ∧ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵)) → 𝐴 ∈ ℝ^*)
15		simpl 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ^* ∧ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵)) → 𝑥 ∈ ℝ^*)
16		xrltle 13145	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝑥 ∈ ℝ^*) → (𝐴 < 𝑥 → 𝐴 ≤ 𝑥))
17	14, 15, 16	syl2anc 593	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ^* ∧ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵)) → (𝐴 < 𝑥 → 𝐴 ≤ 𝑥))
18	17	anim1d 620	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ^* ∧ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵)) → ((𝐴 < 𝑥 ∧ 𝑥 < 𝐵) → (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)))
19	13, 18	anim12d 618	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ^* ∧ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵)) → ((𝑥 ∈ ℚ ∧ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)) → (𝑥 ∈ ℝ^* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵))))
20	19	ex 416	. . . . . . . . . . . 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 484	. . . . . . . . . 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 3171	. . . . . . . 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 1133	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℝ^ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)))
27	9, 26	impbid 214	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (∃𝑥 ∈ ℝ^ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵) ↔ 𝐴 < 𝐵))
28	5, 27	bitrid 285	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (¬ {𝑥 ∈ ℝ^ ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)} = ∅ ↔ 𝐴 < 𝐵))
29		xrltnle 11243	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴))
30	28, 29	bitrd 281	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (¬ {𝑥 ∈ ℝ^ ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)} = ∅ ↔ ¬ 𝐵 ≤ 𝐴))
31	30	con4bid 319	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ({𝑥 ∈ ℝ^ ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < 𝐵)} = ∅ ↔ 𝐵 ≤ 𝐴))
32	2, 31	bitrd 281	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴[,)𝐵) = ∅ ↔ 𝐵 ≤ 𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 399 ∧ w3a 1097 = wceq 1559 ∈ wcel 2141 ≠ wne 2956 ∃wrex 3085 {crab 3413 ∅c0 4283 class class class wbr 5097 (class class class)co 7391 ℝ^*cxr 11209 < clt 11210 ≤ cle 11211 ℚcq 12943 [,)cico 13345

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 7713 ax-cnex 11123 ax-resscn 11124 ax-1cn 11125 ax-icn 11126 ax-addcl 11127 ax-addrcl 11128 ax-mulcl 11129 ax-mulrcl 11130 ax-mulcom 11131 ax-addass 11132 ax-mulass 11133 ax-distr 11134 ax-i2m1 11135 ax-1ne0 11136 ax-1rid 11137 ax-rnegex 11138 ax-rrecex 11139 ax-cnre 11140 ax-pre-lttri 11141 ax-pre-lttrn 11142 ax-pre-ltadd 11143 ax-pre-mulgt0 11144 ax-pre-sup 11145

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-rmo 3366 df-reu 3367 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-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-iun 4948 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-we 5598 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-pred 6283 df-ord 6344 df-on 6345 df-lim 6346 df-suc 6347 df-iota 6472 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-riota 7348 df-ov 7394 df-oprab 7395 df-mpo 7396 df-om 7842 df-1st 7965 df-2nd 7966 df-frecs 8256 df-wrecs 8287 df-recs 8336 df-rdg 8375 df-er 8672 df-en 8922 df-dom 8923 df-sdom 8924 df-sup 9382 df-inf 9383 df-pnf 11212 df-mnf 11213 df-xr 11214 df-ltxr 11215 df-le 11216 df-sub 11410 df-neg 11411 df-div 11839 df-nn 12205 df-n0 12476 df-z 12563 df-uz 12834 df-q 12944 df-ico 13349

This theorem is referenced by: icombl 25614 ioombl 25615 difioo 32945 volico 46518 voliooico 46527 voliccico 46534 ovn0lem 47100 ovnhoilem1 47136 hspmbllem1 47161

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