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Theorem axcontlem3 24594
Description: Lemma for axcont 24604. Given the separation assumption,  B is a subset of  D. (Contributed by Scott Fenton, 18-Jun-2013.)
Hypothesis
Ref Expression
axcontlem3.1  |-  D  =  { p  e.  ( EE `  N )  |  ( U  Btwn  <. Z ,  p >.  \/  p  Btwn  <. Z ,  U >. ) }
Assertion
Ref Expression
axcontlem3  |-  ( ( ( N  e.  NN  /\  ( A  C_  ( EE `  N )  /\  B  C_  ( EE `  N )  /\  A. x  e.  A  A. y  e.  B  x  Btwn  <. Z ,  y
>. ) )  /\  ( Z  e.  ( EE `  N )  /\  U  e.  A  /\  Z  =/= 
U ) )  ->  B  C_  D )
Distinct variable groups:    A, p, x    B, p, x, y    N, p, x, y    U, p, x, y    Z, p, x, y
Allowed substitution hints:    A( y)    D( x, y, p)

Proof of Theorem axcontlem3
StepHypRef Expression
1 simplr2 998 . 2  |-  ( ( ( N  e.  NN  /\  ( A  C_  ( EE `  N )  /\  B  C_  ( EE `  N )  /\  A. x  e.  A  A. y  e.  B  x  Btwn  <. Z ,  y
>. ) )  /\  ( Z  e.  ( EE `  N )  /\  U  e.  A  /\  Z  =/= 
U ) )  ->  B  C_  ( EE `  N ) )
2 simpr2 962 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( A  C_  ( EE `  N )  /\  B  C_  ( EE `  N )  /\  A. x  e.  A  A. y  e.  B  x  Btwn  <. Z ,  y
>. ) )  /\  ( Z  e.  ( EE `  N )  /\  U  e.  A  /\  Z  =/= 
U ) )  ->  U  e.  A )
32anim1i 551 . . . . 5  |-  ( ( ( ( N  e.  NN  /\  ( A 
C_  ( EE `  N )  /\  B  C_  ( EE `  N
)  /\  A. x  e.  A  A. y  e.  B  x  Btwn  <. Z ,  y >. ) )  /\  ( Z  e.  ( EE `  N )  /\  U  e.  A  /\  Z  =/= 
U ) )  /\  p  e.  B )  ->  ( U  e.  A  /\  p  e.  B
) )
4 simplr3 999 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( A  C_  ( EE `  N )  /\  B  C_  ( EE `  N )  /\  A. x  e.  A  A. y  e.  B  x  Btwn  <. Z ,  y
>. ) )  /\  ( Z  e.  ( EE `  N )  /\  U  e.  A  /\  Z  =/= 
U ) )  ->  A. x  e.  A  A. y  e.  B  x  Btwn  <. Z ,  y
>. )
54adantr 451 . . . . 5  |-  ( ( ( ( N  e.  NN  /\  ( A 
C_  ( EE `  N )  /\  B  C_  ( EE `  N
)  /\  A. x  e.  A  A. y  e.  B  x  Btwn  <. Z ,  y >. ) )  /\  ( Z  e.  ( EE `  N )  /\  U  e.  A  /\  Z  =/= 
U ) )  /\  p  e.  B )  ->  A. x  e.  A  A. y  e.  B  x  Btwn  <. Z ,  y
>. )
6 breq1 4026 . . . . . 6  |-  ( x  =  U  ->  (
x  Btwn  <. Z , 
y >. 
<->  U  Btwn  <. Z , 
y >. ) )
7 opeq2 3797 . . . . . . 7  |-  ( y  =  p  ->  <. Z , 
y >.  =  <. Z ,  p >. )
87breq2d 4035 . . . . . 6  |-  ( y  =  p  ->  ( U  Btwn  <. Z ,  y
>. 
<->  U  Btwn  <. Z ,  p >. ) )
96, 8rspc2v 2890 . . . . 5  |-  ( ( U  e.  A  /\  p  e.  B )  ->  ( A. x  e.  A  A. y  e.  B  x  Btwn  <. Z , 
y >.  ->  U  Btwn  <. Z ,  p >. ) )
103, 5, 9sylc 56 . . . 4  |-  ( ( ( ( N  e.  NN  /\  ( A 
C_  ( EE `  N )  /\  B  C_  ( EE `  N
)  /\  A. x  e.  A  A. y  e.  B  x  Btwn  <. Z ,  y >. ) )  /\  ( Z  e.  ( EE `  N )  /\  U  e.  A  /\  Z  =/= 
U ) )  /\  p  e.  B )  ->  U  Btwn  <. Z ,  p >. )
1110orcd 381 . . 3  |-  ( ( ( ( N  e.  NN  /\  ( A 
C_  ( EE `  N )  /\  B  C_  ( EE `  N
)  /\  A. x  e.  A  A. y  e.  B  x  Btwn  <. Z ,  y >. ) )  /\  ( Z  e.  ( EE `  N )  /\  U  e.  A  /\  Z  =/= 
U ) )  /\  p  e.  B )  ->  ( U  Btwn  <. Z ,  p >.  \/  p  Btwn  <. Z ,  U >. ) )
1211ralrimiva 2626 . 2  |-  ( ( ( N  e.  NN  /\  ( A  C_  ( EE `  N )  /\  B  C_  ( EE `  N )  /\  A. x  e.  A  A. y  e.  B  x  Btwn  <. Z ,  y
>. ) )  /\  ( Z  e.  ( EE `  N )  /\  U  e.  A  /\  Z  =/= 
U ) )  ->  A. p  e.  B  ( U  Btwn  <. Z ,  p >.  \/  p  Btwn  <. Z ,  U >. ) )
13 axcontlem3.1 . . . 4  |-  D  =  { p  e.  ( EE `  N )  |  ( U  Btwn  <. Z ,  p >.  \/  p  Btwn  <. Z ,  U >. ) }
1413sseq2i 3203 . . 3  |-  ( B 
C_  D  <->  B  C_  { p  e.  ( EE `  N
)  |  ( U 
Btwn  <. Z ,  p >.  \/  p  Btwn  <. Z ,  U >. ) } )
15 ssrab 3251 . . 3  |-  ( B 
C_  { p  e.  ( EE `  N
)  |  ( U 
Btwn  <. Z ,  p >.  \/  p  Btwn  <. Z ,  U >. ) }  <->  ( B  C_  ( EE `  N
)  /\  A. p  e.  B  ( U  Btwn  <. Z ,  p >.  \/  p  Btwn  <. Z ,  U >. ) ) )
1614, 15bitri 240 . 2  |-  ( B 
C_  D  <->  ( B  C_  ( EE `  N
)  /\  A. p  e.  B  ( U  Btwn  <. Z ,  p >.  \/  p  Btwn  <. Z ,  U >. ) ) )
171, 12, 16sylanbrc 645 1  |-  ( ( ( N  e.  NN  /\  ( A  C_  ( EE `  N )  /\  B  C_  ( EE `  N )  /\  A. x  e.  A  A. y  e.  B  x  Btwn  <. Z ,  y
>. ) )  /\  ( Z  e.  ( EE `  N )  /\  U  e.  A  /\  Z  =/= 
U ) )  ->  B  C_  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   {crab 2547    C_ wss 3152   <.cop 3643   class class class wbr 4023   ` cfv 5255   NNcn 9746   EEcee 24516    Btwn cbtwn 24517
This theorem is referenced by:  axcontlem9  24600  axcontlem10  24601
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024
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