qbtwnxr - Metamath Proof Explorer

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Theorem qbtwnxr 13256

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 13171	. . 3 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		elxr 13171	. . . . 5 ⊢ (𝐵 ∈ ℝ^* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
3		qbtwnre 13255	. . . . . . 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 11411	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → (𝐴 + 1) ∈ ℝ)
7	6	adantr 486	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐴 + 1) ∈ ℝ)
8		ltp1 12083	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → 𝐴 < (𝐴 + 1))
9	8	adantr 486	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → 𝐴 < (𝐴 + 1))
10		qbtwnre 13255	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ (𝐴 + 1) ∈ ℝ ∧ 𝐴 < (𝐴 + 1)) → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < (𝐴 + 1)))
11	5, 7, 9, 10	syl3anc 1398	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → ∃𝑥 ∈ ℚ (𝐴 < 𝑥 ∧ 𝑥 < (𝐴 + 1)))
12		qre 13006	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℚ → 𝑥 ∈ ℝ)
13	12	ltpnfd 13176	. . . . . . . . . . . . 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 3177	. . . . . . . 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 11283	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
23		breq2 5111	. . . . . . . . 9 ⊢ (𝐵 = -∞ → (𝐴 < 𝐵 ↔ 𝐴 < -∞))
24	23	adantl 487	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 = -∞) → (𝐴 < 𝐵 ↔ 𝐴 < -∞))
25		nltmnf 13184	. . . . . . . . . 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 1458	. . . . 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 5110	. . . . . 6 ⊢ (𝐴 = +∞ → (𝐴 < 𝐵 ↔ +∞ < 𝐵))
33	32	adantr 486	. . . . 5 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 ↔ +∞ < 𝐵))
34		pnfnlt 13183	. . . . . . 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 11553	. . . . . . . . . 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 12085	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ → (𝐵 − 1) < 𝐵)
42	41	adantl 487	. . . . . . . . 9 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → (𝐵 − 1) < 𝐵)
43		qbtwnre 13255	. . . . . . . . 9 ⊢ (((𝐵 − 1) ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐵 − 1) < 𝐵) → ∃𝑥 ∈ ℚ ((𝐵 − 1) < 𝑥 ∧ 𝑥 < 𝐵))
44	39, 40, 42, 43	syl3anc 1398	. . . . . . . 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 13179	. . . . . . . . . . . 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 3177	. . . . . . . 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 11236	. . . . . . . . . 10 ⊢ 1 ∈ ℝ
55		mnflt 13178	. . . . . . . . . 10 ⊢ (1 ∈ ℝ → -∞ < 1)
56	54, 55	ax-mp 5	. . . . . . . . 9 ⊢ -∞ < 1
57		breq1 5110	. . . . . . . . 9 ⊢ (𝐴 = -∞ → (𝐴 < 1 ↔ -∞ < 1))
58	56, 57	mpbiri 261	. . . . . . . 8 ⊢ (𝐴 = -∞ → 𝐴 < 1)
59		ltpnf 13175	. . . . . . . . . 10 ⊢ (1 ∈ ℝ → 1 < +∞)
60	54, 59	ax-mp 5	. . . . . . . . 9 ⊢ 1 < +∞
61		breq2 5111	. . . . . . . . 9 ⊢ (𝐵 = +∞ → (1 < 𝐵 ↔ 1 < +∞))
62	60, 61	mpbiri 261	. . . . . . . 8 ⊢ (𝐵 = +∞ → 1 < 𝐵)
63		1z 12652	. . . . . . . . . 10 ⊢ 1 ∈ ℤ
64		zq 13007	. . . . . . . . . 10 ⊢ (1 ∈ ℤ → 1 ∈ ℚ)
65	63, 64	ax-mp 5	. . . . . . . . 9 ⊢ 1 ∈ ℚ
66		breq2 5111	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → (𝐴 < 𝑥 ↔ 𝐴 < 1))
67		breq1 5110	. . . . . . . . . . 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 3579	. . . . . . . . 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 1458	. . . . 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 1457	. . 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 2145 ∃wrex 3088 class class class wbr 5107 (class class class)co 7417 ℝcr 11127 1c1 11129 + caddc 11131 +∞cpnf 11268 -∞cmnf 11269 ℝ^*cxr 11270 < clt 11271 − cmin 11469 ℤcz 12619 ℚcq 13001

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 2215 ax-ext 2734 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7740 ax-cnex 11184 ax-resscn 11185 ax-1cn 11186 ax-icn 11187 ax-addcl 11188 ax-addrcl 11189 ax-mulcl 11190 ax-mulrcl 11191 ax-mulcom 11192 ax-addass 11193 ax-mulass 11194 ax-distr 11195 ax-i2m1 11196 ax-1ne0 11197 ax-1rid 11198 ax-rnegex 11199 ax-rrecex 11200 ax-cnre 11201 ax-pre-lttri 11202 ax-pre-lttrn 11203 ax-pre-ltadd 11204 ax-pre-mulgt0 11205 ax-pre-sup 11206

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 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 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 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 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 7374 df-ov 7420 df-oprab 7421 df-mpo 7422 df-om 7867 df-1st 7990 df-2nd 7991 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-er 8700 df-en 8957 df-dom 8958 df-sdom 8959 df-sup 9416 df-inf 9417 df-pnf 11273 df-mnf 11274 df-xr 11275 df-ltxr 11276 df-le 11277 df-sub 11471 df-neg 11472 df-div 11900 df-nn 12262 df-n0 12533 df-z 12620 df-uz 12892 df-q 13002

This theorem is used by: qextltlem 13258 xralrple 13261 ixxub 13423 ixxlb 13424 ioo0 13427 ico0 13448 ioc0 13449 blssps 24656 blss 24657 blcld 24737 qdensere 25001 tgqioo 25032 dvlip2 26229 lhop2 26249 itgsubst 26283 itg2gt0cn 38432 qinioo 46373 qelioo 46384 qndenserrnbllem 47130

W3C validator