Theorem nqpr 9836

Description: The canonical embedding of the rationals into the reals. (Contributed by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
nqpr	|- ( A e. Q. -> { x | x <Q A } e. P. )

Distinct variable group:   x, A

Proof of Theorem nqpr

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

Step	Hyp	Ref	Expression
1		nsmallnq 9799	. . . . 5 |- ( A e. Q. -> E. x x <Q A )
2		abn0 3954	. . . . 5 |- ( { x | x <Q A } =/= (/) <-> E. x x <Q A )
3	1, 2	sylibr 224	. . . 4 |- ( A e. Q. -> { x | x <Q A } =/= (/) )
4		0pss 4013	. . . 4 |- ( (/) C. { x | x <Q A } <-> { x | x <Q A } =/= (/) )
5	3, 4	sylibr 224	. . 3 |- ( A e. Q. -> (/) C. { x | x <Q A } )
6		ltrelnq 9748	. . . . . . 7 |- <Q C_ ( Q. X. Q. )
7	6	brel 5168	. . . . . 6 |- ( x <Q A -> ( x e. Q. /\ A e. Q. ) )
8	7	simpld 475	. . . . 5 |- ( x <Q A -> x e. Q. )
9	8	abssi 3677	. . . 4 |- { x | x <Q A } C_ Q.
10		ltsonq 9791	. . . . . . 7 |- <Q Or Q.
11	10, 6	soirri 5522	. . . . . 6 |- -. A <Q A
12		breq1 4656	. . . . . . 7 |- ( x = A -> ( x <Q A <-> A <Q A ) )
13	12	elabg 3351	. . . . . 6 |- ( A e. Q. -> ( A e. { x | x <Q A } <-> A <Q A ) )
14	11, 13	mtbiri 317	. . . . 5 |- ( A e. Q. -> -. A e. { x | x <Q A } )
15	14	ancli 574	. . . 4 |- ( A e. Q. -> ( A e. Q. /\ -. A e. { x | x <Q A } ) )
16		ssnelpss 3718	. . . 4 |- ( { x | x <Q A } C_ Q. -> ( ( A e. Q. /\ -. A e. { x | x <Q A } ) -> { x | x <Q A } C. Q. ) )
17	9, 15, 16	mpsyl 68	. . 3 |- ( A e. Q. -> { x | x <Q A } C. Q. )
18	5, 17	jca 554	. 2 |- ( A e. Q. -> ( (/) C. { x | x <Q A } /\ { x | x <Q A } C. Q. ) )
19		vex 3203	. . . . 5 |- y e. _V
20		breq1 4656	. . . . 5 |- ( x = y -> ( x <Q A <-> y <Q A ) )
21	19, 20	elab 3350	. . . 4 |- ( y e. { x | x <Q A } <-> y <Q A )
22	10, 6	sotri 5523	. . . . . . . . 9 |- ( ( z <Q y /\ y <Q A ) -> z <Q A )
23	22	expcom 451	. . . . . . . 8 |- ( y <Q A -> ( z <Q y -> z <Q A ) )
24	23	adantl 482	. . . . . . 7 |- ( ( A e. Q. /\ y <Q A ) -> ( z <Q y -> z <Q A ) )
25		vex 3203	. . . . . . . 8 |- z e. _V
26		breq1 4656	. . . . . . . 8 |- ( x = z -> ( x <Q A <-> z <Q A ) )
27	25, 26	elab 3350	. . . . . . 7 |- ( z e. { x | x <Q A } <-> z <Q A )
28	24, 27	syl6ibr 242	. . . . . 6 |- ( ( A e. Q. /\ y <Q A ) -> ( z <Q y -> z e. { x | x <Q A } ) )
29	28	alrimiv 1855	. . . . 5 |- ( ( A e. Q. /\ y <Q A ) -> A. z ( z <Q y -> z e. { x | x <Q A } ) )
30		ltbtwnnq 9800	. . . . . . . 8 |- ( y <Q A <-> E. z ( y <Q z /\ z <Q A ) )
31	27	anbi2i 730	. . . . . . . . . . 11 |- ( ( y <Q z /\ z e. { x | x <Q A } ) <-> ( y <Q z /\ z <Q A ) )
32	31	biimpri 218	. . . . . . . . . 10 |- ( ( y <Q z /\ z <Q A ) -> ( y <Q z /\ z e. { x | x <Q A } ) )
33	32	ancomd 467	. . . . . . . . 9 |- ( ( y <Q z /\ z <Q A ) -> ( z e. { x | x <Q A } /\ y <Q z ) )
34	33	eximi 1762	. . . . . . . 8 |- ( E. z ( y <Q z /\ z <Q A ) -> E. z ( z e. { x | x <Q A } /\ y <Q z ) )
35	30, 34	sylbi 207	. . . . . . 7 |- ( y <Q A -> E. z ( z e. { x | x <Q A } /\ y <Q z ) )
36	35	adantl 482	. . . . . 6 |- ( ( A e. Q. /\ y <Q A ) -> E. z ( z e. { x | x <Q A } /\ y <Q z ) )
37		df-rex 2918	. . . . . 6 |- ( E. z e. { x | x <Q A } y <Q z <-> E. z ( z e. { x | x <Q A } /\ y <Q z ) )
38	36, 37	sylibr 224	. . . . 5 |- ( ( A e. Q. /\ y <Q A ) -> E. z e. { x | x <Q A } y <Q z )
39	29, 38	jca 554	. . . 4 |- ( ( A e. Q. /\ y <Q A ) -> ( A. z ( z <Q y -> z e. { x | x <Q A } ) /\ E. z e. { x | x <Q A } y <Q z ) )
40	21, 39	sylan2b 492	. . 3 |- ( ( A e. Q. /\ y e. { x | x <Q A } ) -> ( A. z ( z <Q y -> z e. { x | x <Q A } ) /\ E. z e. { x | x <Q A } y <Q z ) )
41	40	ralrimiva 2966	. 2 |- ( A e. Q. -> A. y e. { x | x <Q A } ( A. z ( z <Q y -> z e. { x | x <Q A } ) /\ E. z e. { x | x <Q A } y <Q z ) )
42		elnp 9809	. 2 |- ( { x | x <Q A } e. P. <-> ( ( (/) C. { x | x <Q A } /\ { x | x <Q A } C. Q. ) /\ A. y e. { x | x <Q A } ( A. z ( z <Q y -> z e. { x | x <Q A } ) /\ E. z e. { x | x <Q A } y <Q z ) ) )
43	18, 41, 42	sylanbrc 698	1 |- ( A e. Q. -> { x | x <Q A } e. P. )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   A.wal 1481   E.wex 1704   e. wcel 1990   {cab 2608   =/= wne 2794   A.wral 2912   E.wrex 2913   C_ wss 3574   C. wpss 3575   (/)c0 3915   class class class wbr 4653   Q.cnq 9674   <Q cltq 9680   P.cnp 9681

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  ax-inf2 8538

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  df-np 9803

This theorem is referenced by:  1pr  9837