Theorem prcdnq 9815

Description: A positive real is closed downwards under the positive fractions. Definition 9-3.1 (ii) of [Gleason] p. 121. (Contributed by NM, 25-Feb-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
prcdnq	|- ( ( A e. P. /\ B e. A ) -> ( C <Q B -> C e. A ) )

Proof of Theorem prcdnq

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

Step	Hyp	Ref	Expression
1		ltrelnq 9748	. . . . . . 7 |- <Q C_ ( Q. X. Q. )
2		relxp 5227	. . . . . . 7 |- Rel ( Q. X. Q. )
3		relss 5206	. . . . . . 7 |- ( <Q C_ ( Q. X. Q. ) -> ( Rel ( Q. X. Q. ) -> Rel <Q ) )
4	1, 2, 3	mp2 9	. . . . . 6 |- Rel <Q
5	4	brrelexi 5158	. . . . 5 |- ( C <Q B -> C e. _V )
6		eleq1 2689	. . . . . . . . 9 |- ( x = B -> ( x e. A <-> B e. A ) )
7	6	anbi2d 740	. . . . . . . 8 |- ( x = B -> ( ( A e. P. /\ x e. A ) <-> ( A e. P. /\ B e. A ) ) )
8		breq2 4657	. . . . . . . 8 |- ( x = B -> ( y <Q x <-> y <Q B ) )
9	7, 8	anbi12d 747	. . . . . . 7 |- ( x = B -> ( ( ( A e. P. /\ x e. A ) /\ y <Q x ) <-> ( ( A e. P. /\ B e. A ) /\ y <Q B ) ) )
10	9	imbi1d 331	. . . . . 6 |- ( x = B -> ( ( ( ( A e. P. /\ x e. A ) /\ y <Q x ) -> y e. A ) <-> ( ( ( A e. P. /\ B e. A ) /\ y <Q B ) -> y e. A ) ) )
11		breq1 4656	. . . . . . . 8 |- ( y = C -> ( y <Q B <-> C <Q B ) )
12	11	anbi2d 740	. . . . . . 7 |- ( y = C -> ( ( ( A e. P. /\ B e. A ) /\ y <Q B ) <-> ( ( A e. P. /\ B e. A ) /\ C <Q B ) ) )
13		eleq1 2689	. . . . . . 7 |- ( y = C -> ( y e. A <-> C e. A ) )
14	12, 13	imbi12d 334	. . . . . 6 |- ( y = C -> ( ( ( ( A e. P. /\ B e. A ) /\ y <Q B ) -> y e. A ) <-> ( ( ( A e. P. /\ B e. A ) /\ C <Q B ) -> C e. A ) ) )
15		elnpi 9810	. . . . . . . . . . 11 |- ( A e. P. <-> ( ( A e. _V /\ (/) C. A /\ A C. Q. ) /\ A. x e. A ( A. y ( y <Q x -> y e. A ) /\ E. y e. A x <Q y ) ) )
16	15	simprbi 480	. . . . . . . . . 10 |- ( A e. P. -> A. x e. A ( A. y ( y <Q x -> y e. A ) /\ E. y e. A x <Q y ) )
17	16	r19.21bi 2932	. . . . . . . . 9 |- ( ( A e. P. /\ x e. A ) -> ( A. y ( y <Q x -> y e. A ) /\ E. y e. A x <Q y ) )
18	17	simpld 475	. . . . . . . 8 |- ( ( A e. P. /\ x e. A ) -> A. y ( y <Q x -> y e. A ) )
19	18	19.21bi 2059	. . . . . . 7 |- ( ( A e. P. /\ x e. A ) -> ( y <Q x -> y e. A ) )
20	19	imp 445	. . . . . 6 |- ( ( ( A e. P. /\ x e. A ) /\ y <Q x ) -> y e. A )
21	10, 14, 20	vtocl2g 3270	. . . . 5 |- ( ( B e. A /\ C e. _V ) -> ( ( ( A e. P. /\ B e. A ) /\ C <Q B ) -> C e. A ) )
22	5, 21	sylan2 491	. . . 4 |- ( ( B e. A /\ C <Q B ) -> ( ( ( A e. P. /\ B e. A ) /\ C <Q B ) -> C e. A ) )
23	22	adantll 750	. . 3 |- ( ( ( A e. P. /\ B e. A ) /\ C <Q B ) -> ( ( ( A e. P. /\ B e. A ) /\ C <Q B ) -> C e. A ) )
24	23	pm2.43i 52	. 2 |- ( ( ( A e. P. /\ B e. A ) /\ C <Q B ) -> C e. A )
25	24	ex 450	1 |- ( ( A e. P. /\ B e. A ) -> ( C <Q B -> C e. A ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   C_ wss 3574   C. wpss 3575   (/)c0 3915   class class class wbr 4653   X. cxp 5112   Rel wrel 5119   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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-ltnq 9740  df-np 9803

This theorem is referenced by:  prub  9816  addclprlem1  9838  mulclprlem  9841  distrlem4pr  9848  1idpr  9851  psslinpr  9853  prlem934  9855  ltaddpr  9856  ltexprlem2  9859  ltexprlem3  9860  ltexprlem6  9863  prlem936  9869  reclem2pr  9870  suplem1pr  9874