Theorem posex 5910

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
posex	|- E.x e. RR (0 < x /\ (x < A /\ x < B))

Distinct variable groups:   x,A   x,B

Proof of Theorem posex

Step	Hyp	Ref	Expression
1		df-ne 1590	. . . . . 6 |- (A =/= B <-> -. A = B)
2		posex.1	. . . . . . 7 |- A e. RR
3		posex.2	. . . . . . 7 |- B e. RR
4	2, 3	lttri2 5584	. . . . . 6 |- (A =/= B <-> (A < B \/ B < A))
5	1, 4	bitr3 175	. . . . 5 |- (-. A = B <-> (A < B \/ B < A))
6	5	biimp 151	. . . 4 |- (-. A = B -> (A < B \/ B < A))
7	6	orri 231	. . 3 |- (A = B \/ (A < B \/ B < A))
8		or12 258	. . 3 |- ((A = B \/ (A < B \/ B < A)) <-> (A < B \/ (A = B \/ B < A)))
9	7, 8	mpbi 189	. 2 |- (A < B \/ (A = B \/ B < A))
10		posex.3	. . . . . . . . 9 |- 0 < A
11	2	halfpos 5906	. . . . . . . . 9 |- (0 < A <-> (A / (1 + 1)) < A)
12	10, 11	mpbi 189	. . . . . . . 8 |- (A / (1 + 1)) < A
13		1re 5447	. . . . . . . . . . 11 |- 1 e. RR
14	13, 13	readdcl 5346	. . . . . . . . . 10 |- (1 + 1) e. RR
15		lt01 5692	. . . . . . . . . . . 12 |- 0 < 1
16	13, 13, 15, 15	addgt0i 5613	. . . . . . . . . . 11 |- 0 < (1 + 1)
17	14, 16	gt0ne0i 5629	. . . . . . . . . 10 |- (1 + 1) =/= 0
18	2, 14, 17	redivcl 5800	. . . . . . . . 9 |- (A / (1 + 1)) e. RR
19	18, 2, 3	lttr 5597	. . . . . . . 8 |- (((A / (1 + 1)) < A /\ A < B) -> (A / (1 + 1)) < B)
20	12, 19	mpan 697	. . . . . . 7 |- (A < B -> (A / (1 + 1)) < B)
21	20, 12	jctil 292	. . . . . 6 |- (A < B -> ((A / (1 + 1)) < A /\ (A / (1 + 1)) < B))
22	2, 14, 10, 16	divgt0i 5862	. . . . . 6 |- 0 < (A / (1 + 1))
23	21, 22	jctil 292	. . . . 5 |- (A < B -> (0 < (A / (1 + 1)) /\ ((A / (1 + 1)) < A /\ (A / (1 + 1)) < B)))
24	23, 18	jctil 292	. . . 4 |- (A < B -> ((A / (1 + 1)) e. RR /\ (0 < (A / (1 + 1)) /\ ((A / (1 + 1)) < A /\ (A / (1 + 1)) < B))))
25		breq2 2628	. . . . . 6 |- (x = (A / (1 + 1)) -> (0 < x <-> 0 < (A / (1 + 1))))
26		breq1 2627	. . . . . . 7 |- (x = (A / (1 + 1)) -> (x < A <-> (A / (1 + 1)) < A))
27		breq1 2627	. . . . . . 7 |- (x = (A / (1 + 1)) -> (x < B <-> (A / (1 + 1)) < B))
28	26, 27	anbi12d 630	. . . . . 6 |- (x = (A / (1 + 1)) -> ((x < A /\ x < B) <-> ((A / (1 + 1)) < A /\ (A / (1 + 1)) < B)))
29	25, 28	anbi12d 630	. . . . 5 |- (x = (A / (1 + 1)) -> ((0 < x /\ (x < A /\ x < B)) <-> (0 < (A / (1 + 1)) /\ ((A / (1 + 1)) < A /\ (A / (1 + 1)) < B))))
30	29	rcla4ev 1880	. . . 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)))
31	24, 30	syl 10	. . 3 |- (A < B -> E.x e. RR (0 < x /\ (x < A /\ x < B)))
32		posex.4	. . . . . . . . . 10 |- 0 < B
33	3	halfpos 5906	. . . . . . . . . 10 |- (0 < B <-> (B / (1 + 1)) < B)
34	32, 33	mpbi 189	. . . . . . . . 9 |- (B / (1 + 1)) < B
35		breq2 2628	. . . . . . . . 9 |- (A = B -> ((B / (1 + 1)) < A <-> (B / (1 + 1)) < B))
36	34, 35	mpbiri 194	. . . . . . . 8 |- (A = B -> (B / (1 + 1)) < A)
37	3, 14, 17	redivcl 5800	. . . . . . . . . 10 |- (B / (1 + 1)) e. RR
38	37, 3, 2	lttr 5597	. . . . . . . . 9 |- (((B / (1 + 1)) < B /\ B < A) -> (B / (1 + 1)) < A)
39	34, 38	mpan 697	. . . . . . . 8 |- (B < A -> (B / (1 + 1)) < A)
40	36, 39	jaoi 341	. . . . . . 7 |- ((A = B \/ B < A) -> (B / (1 + 1)) < A)
41	40, 34	jctir 293	. . . . . 6 |- ((A = B \/ B < A) -> ((B / (1 + 1)) < A /\ (B / (1 + 1)) < B))
42	3, 14, 32, 16	divgt0i 5862	. . . . . 6 |- 0 < (B / (1 + 1))
43	41, 42	jctil 292	. . . . 5 |- ((A = B \/ B < A) -> (0 < (B / (1 + 1)) /\ ((B / (1 + 1)) < A /\ (B / (1 + 1)) < B)))
44	43, 37	jctil 292	. . . 4 |- ((A = B \/ B < A) -> ((B / (1 + 1)) e. RR /\ (0 < (B / (1 + 1)) /\ ((B / (1 + 1)) < A /\ (B / (1 + 1)) < B))))
45		breq2 2628	. . . . . 6 |- (x = (B / (1 + 1)) -> (0 < x <-> 0 < (B / (1 + 1))))
46		breq1 2627	. . . . . . 7 |- (x = (B / (1 + 1)) -> (x < A <-> (B / (1 + 1)) < A))
47		breq1 2627	. . . . . . 7 |- (x = (B / (1 + 1)) -> (x < B <-> (B / (1 + 1)) < B))
48	46, 47	anbi12d 630	. . . . . 6 |- (x = (B / (1 + 1)) -> ((x < A /\ x < B) <-> ((B / (1 + 1)) < A /\ (B / (1 + 1)) < B)))
49	45, 48	anbi12d 630	. . . . 5 |- (x = (B / (1 + 1)) -> ((0 < x /\ (x < A /\ x < B)) <-> (0 < (B / (1 + 1)) /\ ((B / (1 + 1)) < A /\ (B / (1 + 1)) < B))))
50	49	rcla4ev 1880	. . . 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)))
51	44, 50	syl 10	. . 3 |- ((A = B \/ B < A) -> E.x e. RR (0 < x /\ (x < A /\ x < B)))
52	31, 51	jaoi 341	. 2 |- ((A < B \/ (A = B \/ B < A)) -> E.x e. RR (0 < x /\ (x < A /\ x < B)))
53	9, 52	ax-mp 7	1 |- E.x e. RR (0 < x /\ (x < A /\ x < B))

Syntax hints:  -. wn 2   \/ wo 222   /\ wa 223   = wceq 958   e. wcel 960   =/= wne 1588  E.wrex 1649   class class class wbr 2624  (class class class)co 3969  RRcr 5245  0cc0 5246  1c1 5247   + caddc 5249   / cdiv 5306   < clt 5498

This theorem is referenced by:  sqrlem20 6693

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-inf2 4634

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-nel 1591  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-1o 4139  df-oadd 4141  df-omul 4142  df-er 4267  df-ec 4269  df-qs 4272  df-en 4374  df-dom 4375  df-sdom 4376  df-ni 5012  df-pli 5013  df-mi 5014  df-lti 5015  df-plpq 5047  df-mpq 5048  df-enq 5049  df-nq 5050  df-plq 5051  df-mq 5052  df-rq 5053  df-ltq 5054  df-1q 5055  df-np 5098  df-1p 5099  df-plp 5100  df-mp 5101  df-ltp 5102  df-plpr 5176  df-mpr 5177  df-enr 5178  df-nr 5179  df-plr 5180  df-mr 5181  df-ltr 5182  df-0r 5183  df-1r 5184  df-m1r 5185  df-c 5252  df-0 5253  df-1 5254  df-i 5255  df-r 5256  df-plus 5257  df-mul 5258  df-lt 5259  df-sub 5368  df-neg 5370  df-pnf 5499  df-mnf 5500  df-xr 5501  df-ltxr 5502  df-le 5503  df-div 5715

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