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Theorem axcontlem3 23384
Description: Lemma for axcont 23394. 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 1031 . 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 995 . . . . . 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 568 . . . . 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 1032 . . . . . 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 465 . . . . 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 4406 . . . . . 6  |-  ( x  =  U  ->  (
x  Btwn  <. Z , 
y >. 
<->  U  Btwn  <. Z , 
y >. ) )
7 opeq2 4171 . . . . . . 7  |-  ( y  =  p  ->  <. Z , 
y >.  =  <. Z ,  p >. )
87breq2d 4415 . . . . . 6  |-  ( y  =  p  ->  ( U  Btwn  <. Z ,  y
>. 
<->  U  Btwn  <. Z ,  p >. ) )
96, 8rspc2v 3186 . . . . 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 392 . . 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 2830 . 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 3492 . . 3  |-  ( B 
C_  D  <->  B  C_  { p  e.  ( EE `  N
)  |  ( U 
Btwn  <. Z ,  p >.  \/  p  Btwn  <. Z ,  U >. ) } )
15 ssrab 3541 . . 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 664 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 368    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648   A.wral 2799   {crab 2803    C_ wss 3439   <.cop 3994   class class class wbr 4403   ` cfv 5529   NNcn 10436   EEcee 23306    Btwn cbtwn 23307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2804  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-br 4404
This theorem is referenced by:  axcontlem9  23390  axcontlem10  23391
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