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Theorem prcdnq 9162
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 9095 . . . . . . 7  |-  <Q  C_  ( Q.  X.  Q. )
2 relxp 4947 . . . . . . 7  |-  Rel  ( Q.  X.  Q. )
3 relss 4927 . . . . . . 7  |-  (  <Q  C_  ( Q.  X.  Q. )  ->  ( Rel  ( Q.  X.  Q. )  ->  Rel  <Q  ) )
41, 2, 3mp2 9 . . . . . 6  |-  Rel  <Q
54brrelexi 4879 . . . . 5  |-  ( C 
<Q  B  ->  C  e. 
_V )
6 eleq1 2503 . . . . . . . . 9  |-  ( x  =  B  ->  (
x  e.  A  <->  B  e.  A ) )
76anbi2d 703 . . . . . . . 8  |-  ( x  =  B  ->  (
( A  e.  P.  /\  x  e.  A )  <-> 
( A  e.  P.  /\  B  e.  A ) ) )
8 breq2 4296 . . . . . . . 8  |-  ( x  =  B  ->  (
y  <Q  x  <->  y  <Q  B ) )
97, 8anbi12d 710 . . . . . . 7  |-  ( x  =  B  ->  (
( ( A  e. 
P.  /\  x  e.  A )  /\  y  <Q  x )  <->  ( ( A  e.  P.  /\  B  e.  A )  /\  y  <Q  B ) ) )
109imbi1d 317 . . . . . 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 4295 . . . . . . . 8  |-  ( y  =  C  ->  (
y  <Q  B  <->  C  <Q  B ) )
1211anbi2d 703 . . . . . . 7  |-  ( y  =  C  ->  (
( ( A  e. 
P.  /\  B  e.  A )  /\  y  <Q  B )  <->  ( ( A  e.  P.  /\  B  e.  A )  /\  C  <Q  B ) ) )
13 eleq1 2503 . . . . . . 7  |-  ( y  =  C  ->  (
y  e.  A  <->  C  e.  A ) )
1412, 13imbi12d 320 . . . . . 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 9157 . . . . . . . . . . 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 464 . . . . . . . . . 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 2814 . . . . . . . . 9  |-  ( ( A  e.  P.  /\  x  e.  A )  ->  ( A. y ( y  <Q  x  ->  y  e.  A )  /\  E. y  e.  A  x 
<Q  y ) )
1817simpld 459 . . . . . . . 8  |-  ( ( A  e.  P.  /\  x  e.  A )  ->  A. y ( y 
<Q  x  ->  y  e.  A ) )
191819.21bi 1804 . . . . . . 7  |-  ( ( A  e.  P.  /\  x  e.  A )  ->  ( y  <Q  x  ->  y  e.  A ) )
2019imp 429 . . . . . 6  |-  ( ( ( A  e.  P.  /\  x  e.  A )  /\  y  <Q  x
)  ->  y  e.  A )
2110, 14, 20vtocl2g 3034 . . . . 5  |-  ( ( B  e.  A  /\  C  e.  _V )  ->  ( ( ( A  e.  P.  /\  B  e.  A )  /\  C  <Q  B )  ->  C  e.  A ) )
225, 21sylan2 474 . . . 4  |-  ( ( B  e.  A  /\  C  <Q  B )  -> 
( ( ( A  e.  P.  /\  B  e.  A )  /\  C  <Q  B )  ->  C  e.  A ) )
2322adantll 713 . . 3  |-  ( ( ( A  e.  P.  /\  B  e.  A )  /\  C  <Q  B )  ->  ( ( ( A  e.  P.  /\  B  e.  A )  /\  C  <Q  B )  ->  C  e.  A
) )
2423pm2.43i 47 . 2  |-  ( ( ( A  e.  P.  /\  B  e.  A )  /\  C  <Q  B )  ->  C  e.  A
)
2524ex 434 1  |-  ( ( A  e.  P.  /\  B  e.  A )  ->  ( C  <Q  B  ->  C  e.  A )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965   A.wal 1367    = wceq 1369    e. wcel 1756   A.wral 2715   E.wrex 2716   _Vcvv 2972    C_ wss 3328    C. wpss 3329   (/)c0 3637   class class class wbr 4292    X. cxp 4838   Rel wrel 4845   Q.cnq 9019    <Q cltq 9025   P.cnp 9026
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-br 4293  df-opab 4351  df-xp 4846  df-rel 4847  df-ltnq 9087  df-np 9150
This theorem is referenced by:  prub  9163  addclprlem1  9185  mulclprlem  9188  distrlem4pr  9195  1idpr  9198  psslinpr  9200  prlem934  9202  ltaddpr  9203  ltexprlem2  9206  ltexprlem3  9207  ltexprlem6  9210  prlem936  9216  reclem2pr  9217  suplem1pr  9221
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