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Theorem axcontlem3 24474
Description: Lemma for axcont 24484. 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 1037 . 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 1001 . . . . . 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 566 . . . . 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 1038 . . . . . 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 463 . . . . 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 4442 . . . . . 6  |-  ( x  =  U  ->  (
x  Btwn  <. Z , 
y >. 
<->  U  Btwn  <. Z , 
y >. ) )
7 opeq2 4204 . . . . . . 7  |-  ( y  =  p  ->  <. Z , 
y >.  =  <. Z ,  p >. )
87breq2d 4451 . . . . . 6  |-  ( y  =  p  ->  ( U  Btwn  <. Z ,  y
>. 
<->  U  Btwn  <. Z ,  p >. ) )
96, 8rspc2v 3216 . . . . 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 60 . . . 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 390 . . 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 2868 . 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 3514 . . 3  |-  ( B 
C_  D  <->  B  C_  { p  e.  ( EE `  N
)  |  ( U 
Btwn  <. Z ,  p >.  \/  p  Btwn  <. Z ,  U >. ) } )
15 ssrab 3564 . . 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 249 . 2  |-  ( B 
C_  D  <->  ( B  C_  ( EE `  N
)  /\  A. p  e.  B  ( U  Btwn  <. Z ,  p >.  \/  p  Btwn  <. Z ,  U >. ) ) )
171, 12, 16sylanbrc 662 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 setvar class
Syntax hints:    -> wi 4    \/ wo 366    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   {crab 2808    C_ wss 3461   <.cop 4022   class class class wbr 4439   ` cfv 5570   NNcn 10531   EEcee 24396    Btwn cbtwn 24397
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440
This theorem is referenced by:  axcontlem9  24480  axcontlem10  24481
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