qbtwnxr - Metamath Proof Explorer

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Theorem qbtwnxr 13360

Description: The rational numbers are dense in ℝ^*: any two extended real numbers have a rational between them. (Contributed by NM, 6-Feb-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)

**Assertion**
Ref	Expression
qbtwnxr	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵

**Proof of Theorem qbtwnxr**
Step	Hyp	Ref	Expression
1		elxr 13275	. . 3 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		elxr 13275	. . . . 5 ⊢ (𝐵 ∈ ℝ^* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
3		qbtwnre 13359	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵))
4	3	3expia 1139	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
5		simpl 488	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → 𝐴 ∈ ℝ)
6		peano2re 11513	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → (𝐴 + 1) ∈ ℝ)
7	6	adantr 486	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐴 + 1) ∈ ℝ)
8		ltp1 12187	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → 𝐴 < (𝐴 + 1))
9	8	adantr 486	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → 𝐴 < (𝐴 + 1))
10		qbtwnre 13359	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ (𝐴 + 1) ∈ ℝ ∧ 𝐴 < (𝐴 + 1)) → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < (𝐴 + 1)))
11	5, 7, 9, 10	syl3anc 1399	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < (𝐴 + 1)))
12		qre 13110	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℚ → 𝑥 ∈ ℝ)
13	12	ltpnfd 13280	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℚ → 𝑥 < +∞)
14	13	adantl 487	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 = +∞) ∧ 𝑥 ∈ ℚ) → 𝑥 < +∞)
15		simplr 781	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 = +∞) ∧ 𝑥 ∈ ℚ) → 𝐵 = +∞)
16	14, 15	breqtrrd 5137	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 = +∞) ∧ 𝑥 ∈ ℚ) → 𝑥 < 𝐵)
17	16	a1d 26	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 = +∞) ∧ 𝑥 ∈ ℚ) → (𝑥 < (𝐴 + 1) → 𝑥 < 𝐵))
18	17	anim2d 624	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 = +∞) ∧ 𝑥 ∈ ℚ) → ((𝐴 < 𝑥 ∧ 𝑥 < (𝐴 + 1)) → (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
19	18	reximdva 3179	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < (𝐴 + 1)) → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
20	11, 19	mpd 16	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵))
21	20	a1d 26	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
22		rexr 11385	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
23		breq2 5110	. . . . . . . . 9 ⊢ (𝐵 = -∞ → (𝐴 < 𝐵 ↔ 𝐴 < -∞))
24	23	adantl 487	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = -∞) → (𝐴 < 𝐵 ↔ 𝐴 < -∞))
25		nltmnf 13288	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ^* → ¬ 𝐴 < -∞)
26	25	adantr 486	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = -∞) → ¬ 𝐴 < -∞)
27	26	pm2.21d 122	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = -∞) → (𝐴 < -∞ → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
28	24, 27	sylbid 243	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = -∞) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
29	22, 28	sylan 592	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
30	4, 21, 29	3jaodan 1459	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
31	2, 30	sylan2b 606	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
32		breq1 5109	. . . . . 6 ⊢ (𝐴 = +∞ → (𝐴 < 𝐵 ↔ +∞ < 𝐵))
33	32	adantr 486	. . . . 5 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 ↔ +∞ < 𝐵))
34		pnfnlt 13287	. . . . . . 7 ⊢ (𝐵 ∈ ℝ^* → ¬ +∞ < 𝐵)
35	34	adantl 487	. . . . . 6 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → ¬ +∞ < 𝐵)
36	35	pm2.21d 122	. . . . 5 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (+∞ < 𝐵 → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
37	33, 36	sylbid 243	. . . 4 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
38		peano2rem 11655	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ → (𝐵 − 1) ∈ ℝ)
39	38	adantl 487	. . . . . . . . 9 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → (𝐵 − 1) ∈ ℝ)
40		simpr 490	. . . . . . . . 9 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ℝ)
41		ltm1 12189	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ → (𝐵 − 1) < 𝐵)
42	41	adantl 487	. . . . . . . . 9 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → (𝐵 − 1) < 𝐵)
43		qbtwnre 13359	. . . . . . . . 9 ⊢ (((𝐵 − 1) ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐵 − 1) < 𝐵) → ∃𝑥 ∈ ℚ ((𝐵 − 1) < 𝑥 ∧ 𝑥 < 𝐵))
44	39, 40, 42, 43	syl3anc 1399	. . . . . . . 8 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → ∃𝑥 ∈ ℚ ((𝐵 − 1) < 𝑥 ∧ 𝑥 < 𝐵))
45		simpll 779	. . . . . . . . . . . 12 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) ∧ 𝑥 ∈ ℚ) → 𝐴 = -∞)
46	12	adantl 487	. . . . . . . . . . . . 13 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) ∧ 𝑥 ∈ ℚ) → 𝑥 ∈ ℝ)
47	46	mnfltd 13283	. . . . . . . . . . . 12 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) ∧ 𝑥 ∈ ℚ) → -∞ < 𝑥)
48	45, 47	eqbrtrd 5131	. . . . . . . . . . 11 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) ∧ 𝑥 ∈ ℚ) → 𝐴 < 𝑥)
49	48	a1d 26	. . . . . . . . . 10 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) ∧ 𝑥 ∈ ℚ) → ((𝐵 − 1) < 𝑥 → 𝐴 < 𝑥))
50	49	anim1d 623	. . . . . . . . 9 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) ∧ 𝑥 ∈ ℚ) → (((𝐵 − 1) < 𝑥 ∧ 𝑥 < 𝐵) → (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
51	50	reximdva 3179	. . . . . . . 8 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → (∃𝑥 ∈ ℚ ((𝐵 − 1) < 𝑥 ∧ 𝑥 < 𝐵) → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
52	44, 51	mpd 16	. . . . . . 7 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵))
53	52	a1d 26	. . . . . 6 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
54		1re 11338	. . . . . . . . . 10 ⊢ 1 ∈ ℝ
55		mnflt 13282	. . . . . . . . . 10 ⊢ (1 ∈ ℝ → -∞ < 1)
56	54, 55	ax-mp 5	. . . . . . . . 9 ⊢ -∞ < 1
57		breq1 5109	. . . . . . . . 9 ⊢ (𝐴 = -∞ → (𝐴 < 1 ↔ -∞ < 1))
58	56, 57	mpbiri 261	. . . . . . . 8 ⊢ (𝐴 = -∞ → 𝐴 < 1)
59		ltpnf 13279	. . . . . . . . . 10 ⊢ (1 ∈ ℝ → 1 < +∞)
60	54, 59	ax-mp 5	. . . . . . . . 9 ⊢ 1 < +∞
61		breq2 5110	. . . . . . . . 9 ⊢ (𝐵 = +∞ → (1 < 𝐵 ↔ 1 < +∞))
62	60, 61	mpbiri 261	. . . . . . . 8 ⊢ (𝐵 = +∞ → 1 < 𝐵)
63		1z 12756	. . . . . . . . . 10 ⊢ 1 ∈ ℤ
64		zq 13111	. . . . . . . . . 10 ⊢ (1 ∈ ℤ → 1 ∈ ℚ)
65	63, 64	ax-mp 5	. . . . . . . . 9 ⊢ 1 ∈ ℚ
66		breq2 5110	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → (𝐴 < 𝑥 ↔ 𝐴 < 1))
67		breq1 5109	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → (𝑥 < 𝐵 ↔ 1 < 𝐵))
68	66, 67	anbi12d 644	. . . . . . . . . 10 ⊢ (𝑥 = 1 → ((𝐴 < 𝑥 ∧ 𝑥 < 𝐵) ↔ (𝐴 < 1 ∧ 1 < 𝐵)))
69	68	rspcev 3580	. . . . . . . . 9 ⊢ ((1 ∈ ℚ ∧ (𝐴 < 1 ∧ 1 < 𝐵)) → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵))
70	65, 69	mpan 703	. . . . . . . 8 ⊢ ((𝐴 < 1 ∧ 1 < 𝐵) → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵))
71	58, 62, 70	syl2an 608	. . . . . . 7 ⊢ ((𝐴 = -∞ ∧ 𝐵 = +∞) → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵))
72	71	a1d 26	. . . . . 6 ⊢ ((𝐴 = -∞ ∧ 𝐵 = +∞) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
73		3mix3 1351	. . . . . . . 8 ⊢ (𝐴 = -∞ → (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
74	73, 1	sylibr 237	. . . . . . 7 ⊢ (𝐴 = -∞ → 𝐴 ∈ ℝ^*)
75	74, 28	sylan 592	. . . . . 6 ⊢ ((𝐴 = -∞ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
76	53, 72, 75	3jaodan 1459	. . . . 5 ⊢ ((𝐴 = -∞ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
77	2, 76	sylan2b 606	. . . 4 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
78	31, 37, 77	3jaoian 1458	. . 3 ⊢ (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
79	1, 78	sylanb 593	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)))
80	79	3impia 1135	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ w3o 1102 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 ∃wrex 3090 class class class wbr 5106 (class class class)co 7446 ℝcr 11229 1c1 11231 + caddc 11233 +∞cpnf 11370 -∞cmnf 11371 ℝ^*cxr 11372 < clt 11373 − cmin 11571 ℤcz 12723 ℚcq 13105

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5253 ax-nul 5264 ax-pow 5332 ax-pr 5396 ax-un 7778 ax-cnex 11286 ax-resscn 11287 ax-1cn 11288 ax-icn 11289 ax-addcl 11290 ax-addrcl 11291 ax-mulcl 11292 ax-mulrcl 11293 ax-mulcom 11294 ax-addass 11295 ax-mulass 11296 ax-distr 11297 ax-i2m1 11298 ax-1ne0 11299 ax-1rid 11300 ax-rnegex 11301 ax-rrecex 11302 ax-cnre 11303 ax-pre-lttri 11304 ax-pre-lttrn 11305 ax-pre-ltadd 11306 ax-pre-mulgt0 11307 ax-pre-sup 11308

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-iun 4956 df-br 5107 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5551 df-eprel 5556 df-po 5564 df-so 5565 df-fr 5609 df-we 5611 df-xp 5662 df-rel 5663 df-cnv 5664 df-co 5665 df-dm 5666 df-rn 5667 df-res 5668 df-ima 5669 df-pred 6325 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6515 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7403 df-ov 7449 df-oprab 7450 df-mpo 7451 df-om 7905 df-1st 8028 df-2nd 8029 df-frecs 8330 df-wrecs 8361 df-recs 8410 df-rdg 8449 df-er 8748 df-en 9005 df-dom 9006 df-sdom 9007 df-sup 9464 df-inf 9465 df-pnf 11375 df-mnf 11376 df-xr 11377 df-ltxr 11378 df-le 11379 df-sub 11573 df-neg 11574 df-div 12004 df-nn 12366 df-n0 12637 df-z 12724 df-uz 12996 df-q 13106

This theorem is used by: qextltlem 13362 xralrple 13365 ixxub 13527 ixxlb 13528 ioo0 13531 ico0 13552 ioc0 13553 blssps 24776 blss 24777 blcld 24857 qdensere 25121 tgqioo 25152 dvlip2 26348 lhop2 26368 itgsubst 26402 itg2gt0cn 38798 qinioo 46762 qelioo 46773 qndenserrnbllem 47519

W3C validator