Theorem posexi 7091

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

Hypotheses

Ref	Expression
posex.1	|- A e. RR
posex.2	|- B e. RR
posex.3	|- 0 < A
posex.4	|- 0 < B

Assertion

Ref	Expression
posexi	|- E.x e. RR (0 < x /\ (x < A /\ x < B))

Distinct variable groups:   x,A   x,B

Proof of Theorem posexi

Step	Hyp	Ref	Expression
1		df-ne 2019	. . . . . 6 |- (A =/= B <-> -. A = B)
2		posex.1	. . . . . . 7 |- A e. RR
3		posex.2	. . . . . . 7 |- B e. RR
4	2, 3	lttri2i 6747	. . . . . 6 |- (A =/= B <-> (A < B \/ B < A))
5	1, 4	bitr3i 192	. . . . 5 |- (-. A = B <-> (A < B \/ B < A))
6	5	biimpi 168	. . . 4 |- (-. A = B -> (A < B \/ B < A))
7	6	orri 248	. . 3 |- (A = B \/ (A < B \/ B < A))
8		or12 278	. . 3 |- ((A = B \/ (A < B \/ B < A)) <-> (A < B \/ (A = B \/ B < A)))
9	7, 8	mpbi 206	. 2 |- (A < B \/ (A = B \/ B < A))
10		breq2 3342	. . . . . 6 |- (x = (A / (1 + 1)) -> (0 < x <-> 0 < (A / (1 + 1))))
11		breq1 3341	. . . . . . 7 |- (x = (A / (1 + 1)) -> (x < A <-> (A / (1 + 1)) < A))
12		breq1 3341	. . . . . . 7 |- (x = (A / (1 + 1)) -> (x < B <-> (A / (1 + 1)) < B))
13	11, 12	anbi12d 690	. . . . . 6 |- (x = (A / (1 + 1)) -> ((x < A /\ x < B) <-> ((A / (1 + 1)) < A /\ (A / (1 + 1)) < B)))
14	10, 13	anbi12d 690	. . . . 5 |- (x = (A / (1 + 1)) -> ((0 < x /\ (x < A /\ x < B)) <-> (0 < (A / (1 + 1)) /\ ((A / (1 + 1)) < A /\ (A / (1 + 1)) < B))))
15	14	rcla4ev 2381	. . . 4 |- (((A / (1 + 1)) e. RR /\ (0 < (A / (1 + 1)) /\ ((A / (1 + 1)) < A /\ (A / (1 + 1)) < B))) -> E.x e. RR (0 < x /\ (x < A /\ x < B)))
16		1re 6598	. . . . . 6 |- 1 e. RR
17	16, 16	readdcli 6487	. . . . 5 |- (1 + 1) e. RR
18		lt01 6871	. . . . . . 7 |- 0 < 1
19	16, 16, 18, 18	addgt0ii 6781	. . . . . 6 |- 0 < (1 + 1)
20	17, 19	gt0ne0ii 6799	. . . . 5 |- (1 + 1) =/= 0
21	2, 17, 20	redivcli 6976	. . . 4 |- (A / (1 + 1)) e. RR
22		posex.3	. . . . . . . 8 |- 0 < A
23	2	halfposi 7087	. . . . . . . 8 |- (0 < A <-> (A / (1 + 1)) < A)
24	22, 23	mpbi 206	. . . . . . 7 |- (A / (1 + 1)) < A
25	21, 2, 3	lttri 6760	. . . . . . 7 |- (((A / (1 + 1)) < A /\ A < B) -> (A / (1 + 1)) < B)
26	24, 25	mpan 759	. . . . . 6 |- (A < B -> (A / (1 + 1)) < B)
27	26, 24	jctil 316	. . . . 5 |- (A < B -> ((A / (1 + 1)) < A /\ (A / (1 + 1)) < B))
28	2, 17, 22, 19	divgt0ii 7042	. . . . 5 |- 0 < (A / (1 + 1))
29	27, 28	jctil 316	. . . 4 |- (A < B -> (0 < (A / (1 + 1)) /\ ((A / (1 + 1)) < A /\ (A / (1 + 1)) < B)))
30	15, 21, 29	sylancr 526	. . 3 |- (A < B -> E.x e. RR (0 < x /\ (x < A /\ x < B)))
31		breq2 3342	. . . . . 6 |- (x = (B / (1 + 1)) -> (0 < x <-> 0 < (B / (1 + 1))))
32		breq1 3341	. . . . . . 7 |- (x = (B / (1 + 1)) -> (x < A <-> (B / (1 + 1)) < A))
33		breq1 3341	. . . . . . 7 |- (x = (B / (1 + 1)) -> (x < B <-> (B / (1 + 1)) < B))
34	32, 33	anbi12d 690	. . . . . 6 |- (x = (B / (1 + 1)) -> ((x < A /\ x < B) <-> ((B / (1 + 1)) < A /\ (B / (1 + 1)) < B)))
35	31, 34	anbi12d 690	. . . . 5 |- (x = (B / (1 + 1)) -> ((0 < x /\ (x < A /\ x < B)) <-> (0 < (B / (1 + 1)) /\ ((B / (1 + 1)) < A /\ (B / (1 + 1)) < B))))
36	35	rcla4ev 2381	. . . 4 |- (((B / (1 + 1)) e. RR /\ (0 < (B / (1 + 1)) /\ ((B / (1 + 1)) < A /\ (B / (1 + 1)) < B))) -> E.x e. RR (0 < x /\ (x < A /\ x < B)))
37	3, 17, 20	redivcli 6976	. . . 4 |- (B / (1 + 1)) e. RR
38		posex.4	. . . . . . . . 9 |- 0 < B
39	3	halfposi 7087	. . . . . . . . 9 |- (0 < B <-> (B / (1 + 1)) < B)
40	38, 39	mpbi 206	. . . . . . . 8 |- (B / (1 + 1)) < B
41		breq2 3342	. . . . . . . 8 |- (A = B -> ((B / (1 + 1)) < A <-> (B / (1 + 1)) < B))
42	40, 41	mpbiri 211	. . . . . . 7 |- (A = B -> (B / (1 + 1)) < A)
43	37, 3, 2	lttri 6760	. . . . . . . 8 |- (((B / (1 + 1)) < B /\ B < A) -> (B / (1 + 1)) < A)
44	40, 43	mpan 759	. . . . . . 7 |- (B < A -> (B / (1 + 1)) < A)
45	42, 44	jaoi 368	. . . . . 6 |- ((A = B \/ B < A) -> (B / (1 + 1)) < A)
46	45, 40	jctir 317	. . . . 5 |- ((A = B \/ B < A) -> ((B / (1 + 1)) < A /\ (B / (1 + 1)) < B))
47	3, 17, 38, 19	divgt0ii 7042	. . . . 5 |- 0 < (B / (1 + 1))
48	46, 47	jctil 316	. . . 4 |- ((A = B \/ B < A) -> (0 < (B / (1 + 1)) /\ ((B / (1 + 1)) < A /\ (B / (1 + 1)) < B)))
49	36, 37, 48	sylancr 526	. . 3 |- ((A = B \/ B < A) -> E.x e. RR (0 < x /\ (x < A /\ x < B)))
50	30, 49	jaoi 368	. 2 |- ((A < B \/ (A = B \/ B < A)) -> E.x e. RR (0 < x /\ (x < A /\ x < B)))
51	9, 50	ax-mp 7	1 |- E.x e. RR (0 < x /\ (x < A /\ x < B))

Syntax hints:  -. wn 2   \/ wo 239   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017  E.wrex 2106   class class class wbr 3338  (class class class)co 4884  RRcr 6385  0cc0 6386  1c1 6387   + caddc 6389   / cdiv 6447   < clt 6653

This theorem is referenced by:  sqrlem20 7942

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892

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