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Theorem prcdnq 8872
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.
StepHypRef Expression
1 ltrelnq 8805 . . . . . . 7  |-  <Q  C_  ( Q.  X.  Q. )
2 relxp 4985 . . . . . . 7  |-  Rel  ( Q.  X.  Q. )
3 relss 4965 . . . . . . 7  |-  (  <Q  C_  ( Q.  X.  Q. )  ->  ( Rel  ( Q.  X.  Q. )  ->  Rel  <Q  ) )
41, 2, 3mp2 9 . . . . . 6  |-  Rel  <Q
54brrelexi 4920 . . . . 5  |-  ( C 
<Q  B  ->  C  e. 
_V )
6 eleq1 2498 . . . . . . . . 9  |-  ( x  =  B  ->  (
x  e.  A  <->  B  e.  A ) )
76anbi2d 686 . . . . . . . 8  |-  ( x  =  B  ->  (
( A  e.  P.  /\  x  e.  A )  <-> 
( A  e.  P.  /\  B  e.  A ) ) )
8 breq2 4218 . . . . . . . 8  |-  ( x  =  B  ->  (
y  <Q  x  <->  y  <Q  B ) )
97, 8anbi12d 693 . . . . . . 7  |-  ( x  =  B  ->  (
( ( A  e. 
P.  /\  x  e.  A )  /\  y  <Q  x )  <->  ( ( A  e.  P.  /\  B  e.  A )  /\  y  <Q  B ) ) )
109imbi1d 310 . . . . . 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 4217 . . . . . . . 8  |-  ( y  =  C  ->  (
y  <Q  B  <->  C  <Q  B ) )
1211anbi2d 686 . . . . . . 7  |-  ( y  =  C  ->  (
( ( A  e. 
P.  /\  B  e.  A )  /\  y  <Q  B )  <->  ( ( A  e.  P.  /\  B  e.  A )  /\  C  <Q  B ) ) )
13 eleq1 2498 . . . . . . 7  |-  ( y  =  C  ->  (
y  e.  A  <->  C  e.  A ) )
1412, 13imbi12d 313 . . . . . 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 8867 . . . . . . . . . . 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 ) ) )
1615simprbi 452 . . . . . . . . . 10  |-  ( A  e.  P.  ->  A. x  e.  A  ( A. y ( y  <Q  x  ->  y  e.  A
)  /\  E. y  e.  A  x  <Q  y ) )
1716r19.21bi 2806 . . . . . . . . 9  |-  ( ( A  e.  P.  /\  x  e.  A )  ->  ( A. y ( y  <Q  x  ->  y  e.  A )  /\  E. y  e.  A  x 
<Q  y ) )
1817simpld 447 . . . . . . . 8  |-  ( ( A  e.  P.  /\  x  e.  A )  ->  A. y ( y 
<Q  x  ->  y  e.  A ) )
191819.21bi 1775 . . . . . . 7  |-  ( ( A  e.  P.  /\  x  e.  A )  ->  ( y  <Q  x  ->  y  e.  A ) )
2019imp 420 . . . . . 6  |-  ( ( ( A  e.  P.  /\  x  e.  A )  /\  y  <Q  x
)  ->  y  e.  A )
2110, 14, 20vtocl2g 3017 . . . . 5  |-  ( ( B  e.  A  /\  C  e.  _V )  ->  ( ( ( A  e.  P.  /\  B  e.  A )  /\  C  <Q  B )  ->  C  e.  A ) )
225, 21sylan2 462 . . . 4  |-  ( ( B  e.  A  /\  C  <Q  B )  -> 
( ( ( A  e.  P.  /\  B  e.  A )  /\  C  <Q  B )  ->  C  e.  A ) )
2322adantll 696 . . 3  |-  ( ( ( A  e.  P.  /\  B  e.  A )  /\  C  <Q  B )  ->  ( ( ( A  e.  P.  /\  B  e.  A )  /\  C  <Q  B )  ->  C  e.  A
) )
2423pm2.43i 46 . 2  |-  ( ( ( A  e.  P.  /\  B  e.  A )  /\  C  <Q  B )  ->  C  e.  A
)
2524ex 425 1  |-  ( ( A  e.  P.  /\  B  e.  A )  ->  ( C  <Q  B  ->  C  e.  A )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937   A.wal 1550    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708   _Vcvv 2958    C_ wss 3322    C. wpss 3323   (/)c0 3630   class class class wbr 4214    X. cxp 4878   Rel wrel 4885   Q.cnq 8729    <Q cltq 8735   P.cnp 8736
This theorem is referenced by:  prub  8873  addclprlem1  8895  mulclprlem  8898  distrlem4pr  8905  1idpr  8908  psslinpr  8910  prlem934  8912  ltaddpr  8913  ltexprlem2  8916  ltexprlem3  8917  ltexprlem6  8920  prlem936  8926  reclem2pr  8927  suplem1pr  8931
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215  df-opab 4269  df-xp 4886  df-rel 4887  df-ltnq 8797  df-np 8860
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