Theorem ltbtwnnq 9800

Description: There exists a number between any two positive fractions. Proposition 9-2.6(i) of [Gleason] p. 120. (Contributed by NM, 17-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
ltbtwnnq	|- ( A <Q B <-> E. x ( A <Q x /\ x <Q B ) )

Distinct variable groups:   x, A   x, B

Proof of Theorem ltbtwnnq

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltrelnq 9748	. . . . 5 |- <Q C_ ( Q. X. Q. )
2	1	brel 5168	. . . 4 |- ( A <Q B -> ( A e. Q. /\ B e. Q. ) )
3	2	simprd 479	. . 3 |- ( A <Q B -> B e. Q. )
4		ltexnq 9797	. . . 4 |- ( B e. Q. -> ( A <Q B <-> E. y ( A +Q y ) = B ) )
5		eleq1 2689	. . . . . . . . . 10 |- ( ( A +Q y ) = B -> ( ( A +Q y ) e. Q. <-> B e. Q. ) )
6	5	biimparc 504	. . . . . . . . 9 |- ( ( B e. Q. /\ ( A +Q y ) = B ) -> ( A +Q y ) e. Q. )
7		addnqf 9770	. . . . . . . . . . 11 |- +Q : ( Q. X. Q. ) --> Q.
8	7	fdmi 6052	. . . . . . . . . 10 |- dom +Q = ( Q. X. Q. )
9		0nnq 9746	. . . . . . . . . 10 |- -. (/) e. Q.
10	8, 9	ndmovrcl 6820	. . . . . . . . 9 |- ( ( A +Q y ) e. Q. -> ( A e. Q. /\ y e. Q. ) )
11	6, 10	syl 17	. . . . . . . 8 |- ( ( B e. Q. /\ ( A +Q y ) = B ) -> ( A e. Q. /\ y e. Q. ) )
12	11	simprd 479	. . . . . . 7 |- ( ( B e. Q. /\ ( A +Q y ) = B ) -> y e. Q. )
13		nsmallnq 9799	. . . . . . . 8 |- ( y e. Q. -> E. z z <Q y )
14	11	simpld 475	. . . . . . . . . . . 12 |- ( ( B e. Q. /\ ( A +Q y ) = B ) -> A e. Q. )
15	1	brel 5168	. . . . . . . . . . . . 13 |- ( z <Q y -> ( z e. Q. /\ y e. Q. ) )
16	15	simpld 475	. . . . . . . . . . . 12 |- ( z <Q y -> z e. Q. )
17		ltaddnq 9796	. . . . . . . . . . . 12 |- ( ( A e. Q. /\ z e. Q. ) -> A <Q ( A +Q z ) )
18	14, 16, 17	syl2an 494	. . . . . . . . . . 11 |- ( ( ( B e. Q. /\ ( A +Q y ) = B ) /\ z <Q y ) -> A <Q ( A +Q z ) )
19		ltanq 9793	. . . . . . . . . . . . . 14 |- ( A e. Q. -> ( z <Q y <-> ( A +Q z ) <Q ( A +Q y ) ) )
20	19	biimpa 501	. . . . . . . . . . . . 13 |- ( ( A e. Q. /\ z <Q y ) -> ( A +Q z ) <Q ( A +Q y ) )
21	14, 20	sylan 488	. . . . . . . . . . . 12 |- ( ( ( B e. Q. /\ ( A +Q y ) = B ) /\ z <Q y ) -> ( A +Q z ) <Q ( A +Q y ) )
22		simplr 792	. . . . . . . . . . . 12 |- ( ( ( B e. Q. /\ ( A +Q y ) = B ) /\ z <Q y ) -> ( A +Q y ) = B )
23	21, 22	breqtrd 4679	. . . . . . . . . . 11 |- ( ( ( B e. Q. /\ ( A +Q y ) = B ) /\ z <Q y ) -> ( A +Q z ) <Q B )
24		ovex 6678	. . . . . . . . . . . 12 |- ( A +Q z ) e. _V
25		breq2 4657	. . . . . . . . . . . . 13 |- ( x = ( A +Q z ) -> ( A <Q x <-> A <Q ( A +Q z ) ) )
26		breq1 4656	. . . . . . . . . . . . 13 |- ( x = ( A +Q z ) -> ( x <Q B <-> ( A +Q z ) <Q B ) )
27	25, 26	anbi12d 747	. . . . . . . . . . . 12 |- ( x = ( A +Q z ) -> ( ( A <Q x /\ x <Q B ) <-> ( A <Q ( A +Q z ) /\ ( A +Q z ) <Q B ) ) )
28	24, 27	spcev 3300	. . . . . . . . . . 11 |- ( ( A <Q ( A +Q z ) /\ ( A +Q z ) <Q B ) -> E. x ( A <Q x /\ x <Q B ) )
29	18, 23, 28	syl2anc 693	. . . . . . . . . 10 |- ( ( ( B e. Q. /\ ( A +Q y ) = B ) /\ z <Q y ) -> E. x ( A <Q x /\ x <Q B ) )
30	29	ex 450	. . . . . . . . 9 |- ( ( B e. Q. /\ ( A +Q y ) = B ) -> ( z <Q y -> E. x ( A <Q x /\ x <Q B ) ) )
31	30	exlimdv 1861	. . . . . . . 8 |- ( ( B e. Q. /\ ( A +Q y ) = B ) -> ( E. z z <Q y -> E. x ( A <Q x /\ x <Q B ) ) )
32	13, 31	syl5 34	. . . . . . 7 |- ( ( B e. Q. /\ ( A +Q y ) = B ) -> ( y e. Q. -> E. x ( A <Q x /\ x <Q B ) ) )
33	12, 32	mpd 15	. . . . . 6 |- ( ( B e. Q. /\ ( A +Q y ) = B ) -> E. x ( A <Q x /\ x <Q B ) )
34	33	ex 450	. . . . 5 |- ( B e. Q. -> ( ( A +Q y ) = B -> E. x ( A <Q x /\ x <Q B ) ) )
35	34	exlimdv 1861	. . . 4 |- ( B e. Q. -> ( E. y ( A +Q y ) = B -> E. x ( A <Q x /\ x <Q B ) ) )
36	4, 35	sylbid 230	. . 3 |- ( B e. Q. -> ( A <Q B -> E. x ( A <Q x /\ x <Q B ) ) )
37	3, 36	mpcom 38	. 2 |- ( A <Q B -> E. x ( A <Q x /\ x <Q B ) )
38		ltsonq 9791	. . . 4 |- <Q Or Q.
39	38, 1	sotri 5523	. . 3 |- ( ( A <Q x /\ x <Q B ) -> A <Q B )
40	39	exlimiv 1858	. 2 |- ( E. x ( A <Q x /\ x <Q B ) -> A <Q B )
41	37, 40	impbii 199	1 |- ( A <Q B <-> E. x ( A <Q x /\ x <Q B ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   class class class wbr 4653   X. cxp 5112  (class class class)co 6650   Q.cnq 9674   +Q cplq 9677   <Q cltq 9680

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-omul 7565  df-er 7742  df-ni 9694  df-pli 9695  df-mi 9696  df-lti 9697  df-plpq 9730  df-mpq 9731  df-ltpq 9732  df-enq 9733  df-nq 9734  df-erq 9735  df-plq 9736  df-mq 9737  df-1nq 9738  df-rq 9739  df-ltnq 9740

This theorem is referenced by:  nqpr  9836  reclem2pr  9870