Theorem posex 4422

Description: There exists a positive number less than two others.

Hypotheses

Ref	Expression
posex.1	⊢ A ∈ ℝ
posex.2	⊢ B ∈ ℝ
posex.3	⊢ 0 < A
posex.4	⊢ 0 < B

Assertion

Ref	Expression
posex	⊢ ∃x ∈ ℝ (0 < x ∧ (x < A ∧ x < B))

Distinct variable group(s):   x,A   x,B

Proof of Theorem posex

Step	Hyp	Ref	Expression
1		posex.1	. . . . . 6 ⊢ A ∈ ℝ
2		posex.2	. . . . . 6 ⊢ B ∈ ℝ
3	1, 2	lttri2 4295	. . . . 5 ⊢ (¬ A = B ↔ (A < B ∨ B < A))
4	3	biimp 133	. . . 4 ⊢ (¬ A = B → (A < B ∨ B < A))
5	4	orri 201	. . 3 ⊢ (A = B ∨ (A < B ∨ B < A))
6		or12 217	. . 3 ⊢ ((A = B ∨ (A < B ∨ B < A)) ↔ (A < B ∨ (A = B ∨ B < A)))
7	5, 6	mpbi 164	. 2 ⊢ (A < B ∨ (A = B ∨ B < A))
8		posex.3	. . . . . . . . 9 ⊢ 0 < A
9	1	halfpos 4421	. . . . . . . . 9 ⊢ (0 < A ↔ (A / (1 + 1)) < A)
10	8, 9	mpbi 164	. . . . . . . 8 ⊢ (A / (1 + 1)) < A
11		ax1re 4064	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ
12	11, 11	readdcl 4118	. . . . . . . . . 10 ⊢ (1 + 1) ∈ ℝ
13		lt01 4377	. . . . . . . . . . . 12 ⊢ 0 < 1
14	11, 11, 13, 13	addgt0i 4326	. . . . . . . . . . 11 ⊢ 0 < (1 + 1)
15	12, 14	gt0ne0i 4345	. . . . . . . . . 10 ⊢ (1 + 1) ≠ 0
16	1, 12, 15	redivcl 4274	. . . . . . . . 9 ⊢ (A / (1 + 1)) ∈ ℝ
17	16, 1, 2	lttr 4307	. . . . . . . 8 ⊢ (((A / (1 + 1)) < A ∧ A < B) → (A / (1 + 1)) < B)
18	10, 17	mpan 518	. . . . . . 7 ⊢ (A < B → (A / (1 + 1)) < B)
19	18, 10	jctil 240	. . . . . 6 ⊢ (A < B → ((A / (1 + 1)) < A ∧ (A / (1 + 1)) < B))
20	1, 12, 8, 14	divgt0i 4391	. . . . . 6 ⊢ 0 < (A / (1 + 1))
21	19, 20	jctil 240	. . . . 5 ⊢ (A < B → (0 < (A / (1 + 1)) ∧ ((A / (1 + 1)) < A ∧ (A / (1 + 1)) < B)))
22	21, 16	jctil 240	. . . 4 ⊢ (A < B → ((A / (1 + 1)) ∈ ℝ ∧ (0 < (A / (1 + 1)) ∧ ((A / (1 + 1)) < A ∧ (A / (1 + 1)) < B))))
23		breq2 2066	. . . . . 6 ⊢ (x = (A / (1 + 1)) → (0 < x ↔ 0 < (A / (1 + 1))))
24		breq1 2065	. . . . . . 7 ⊢ (x = (A / (1 + 1)) → (x < A ↔ (A / (1 + 1)) < A))
25		breq1 2065	. . . . . . 7 ⊢ (x = (A / (1 + 1)) → (x < B ↔ (A / (1 + 1)) < B))
26	24, 25	anbi12d 476	. . . . . 6 ⊢ (x = (A / (1 + 1)) → ((x < A ∧ x < B) ↔ ((A / (1 + 1)) < A ∧ (A / (1 + 1)) < B)))
27	23, 26	anbi12d 476	. . . . 5 ⊢ (x = (A / (1 + 1)) → ((0 < x ∧ (x < A ∧ x < B)) ↔ (0 < (A / (1 + 1)) ∧ ((A / (1 + 1)) < A ∧ (A / (1 + 1)) < B))))
28	27	rcla4ev 1403	. . . 4 ⊢ (((A / (1 + 1)) ∈ ℝ ∧ (0 < (A / (1 + 1)) ∧ ((A / (1 + 1)) < A ∧ (A / (1 + 1)) < B))) → ∃x ∈ ℝ (0 < x ∧ (x < A ∧ x < B)))
29	22, 28	syl 12	. . 3 ⊢ (A < B → ∃x ∈ ℝ (0 < x ∧ (x < A ∧ x < B)))
30		posex.4	. . . . . . . . . 10 ⊢ 0 < B
31	2	halfpos 4421	. . . . . . . . . 10 ⊢ (0 < B ↔ (B / (1 + 1)) < B)
32	30, 31	mpbi 164	. . . . . . . . 9 ⊢ (B / (1 + 1)) < B
33		breq2 2066	. . . . . . . . 9 ⊢ (A = B → ((B / (1 + 1)) < A ↔ (B / (1 + 1)) < B))
34	32, 33	mpbiri 169	. . . . . . . 8 ⊢ (A = B → (B / (1 + 1)) < A)
35	2, 12, 15	redivcl 4274	. . . . . . . . . 10 ⊢ (B / (1 + 1)) ∈ ℝ
36	35, 2, 1	lttr 4307	. . . . . . . . 9 ⊢ (((B / (1 + 1)) < B ∧ B < A) → (B / (1 + 1)) < A)
37	32, 36	mpan 518	. . . . . . . 8 ⊢ (B < A → (B / (1 + 1)) < A)
38	34, 37	jaoi 275	. . . . . . 7 ⊢ ((A = B ∨ B < A) → (B / (1 + 1)) < A)
39	38, 32	jctir 241	. . . . . 6 ⊢ ((A = B ∨ B < A) → ((B / (1 + 1)) < A ∧ (B / (1 + 1)) < B))
40	2, 12, 30, 14	divgt0i 4391	. . . . . 6 ⊢ 0 < (B / (1 + 1))
41	39, 40	jctil 240	. . . . 5 ⊢ ((A = B ∨ B < A) → (0 < (B / (1 + 1)) ∧ ((B / (1 + 1)) < A ∧ (B / (1 + 1)) < B)))
42	41, 35	jctil 240	. . . 4 ⊢ ((A = B ∨ B < A) → ((B / (1 + 1)) ∈ ℝ ∧ (0 < (B / (1 + 1)) ∧ ((B / (1 + 1)) < A ∧ (B / (1 + 1)) < B))))
43		breq2 2066	. . . . . 6 ⊢ (x = (B / (1 + 1)) → (0 < x ↔ 0 < (B / (1 + 1))))
44		breq1 2065	. . . . . . 7 ⊢ (x = (B / (1 + 1)) → (x < A ↔ (B / (1 + 1)) < A))
45		breq1 2065	. . . . . . 7 ⊢ (x = (B / (1 + 1)) → (x < B ↔ (B / (1 + 1)) < B))
46	44, 45	anbi12d 476	. . . . . 6 ⊢ (x = (B / (1 + 1)) → ((x < A ∧ x < B) ↔ ((B / (1 + 1)) < A ∧ (B / (1 + 1)) < B)))
47	43, 46	anbi12d 476	. . . . 5 ⊢ (x = (B / (1 + 1)) → ((0 < x ∧ (x < A ∧ x < B)) ↔ (0 < (B / (1 + 1)) ∧ ((B / (1 + 1)) < A ∧ (B / (1 + 1)) < B))))
48	47	rcla4ev 1403	. . . 4 ⊢ (((B / (1 + 1)) ∈ ℝ ∧ (0 < (B / (1 + 1)) ∧ ((B / (1 + 1)) < A ∧ (B / (1 + 1)) < B))) → ∃x ∈ ℝ (0 < x ∧ (x < A ∧ x < B)))
49	42, 48	syl 12	. . 3 ⊢ ((A = B ∨ B < A) → ∃x ∈ ℝ (0 < x ∧ (x < A ∧ x < B)))
50	29, 49	jaoi 275	. 2 ⊢ ((A < B ∨ (A = B ∨ B < A)) → ∃x ∈ ℝ (0 < x ∧ (x < A ∧ x < B)))
51	7, 50	ax-mp 6	1 ⊢ ∃x ∈ ℝ (0 < x ∧ (x < A ∧ x < B))

Syntax hints:  ¬ wn 1   ∨ wo 195   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ∃wrex 1202   class class class wbr 2054  (class class class)co 3001  ℝcr 4027  0cc0 4028  1c1 4029   + caddc 4031   < clt 4033   / cdiv 4091

This theorem is referenced by:  sqrlem20 4750

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1o 3104  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-ni 3794  df-pli 3795  df-mi 3796  df-lti 3797  df-plpq 3829  df-mpq 3830  df-enq 3831  df-nq 3832  df-plq 3833  df-mq 3834  df-rq 3835  df-ltq 3836  df-1q 3837  df-np 3880  df-1p 3881  df-plp 3882  df-mp 3883  df-ltp 3884  df-plpr 3958  df-mpr 3959  df-enr 3960  df-nr 3961  df-plr 3962  df-mr 3963  df-ltr 3964  df-0r 3965  df-1r 3966  df-m1r 3967  df-c 4034  df-0 4035  df-1 4036  df-r 4038  df-plus 4039  df-mul 4040  df-lt 4041  df-sub 4133  df-neg 4135  df-div 4216  df-le 4277

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