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Theorem axcontlem3 25336
Description: Lemma for axcont 25346. 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 999 . 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 963 . . . . . 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 1000 . . . . . 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 4128 . . . . . 6  |-  ( x  =  U  ->  (
x  Btwn  <. Z , 
y >. 
<->  U  Btwn  <. Z , 
y >. ) )
7 opeq2 3899 . . . . . . 7  |-  ( y  =  p  ->  <. Z , 
y >.  =  <. Z ,  p >. )
87breq2d 4137 . . . . . 6  |-  ( y  =  p  ->  ( U  Btwn  <. Z ,  y
>. 
<->  U  Btwn  <. Z ,  p >. ) )
96, 8rspc2v 2975 . . . . 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 2711 . 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 3289 . . 3  |-  ( B 
C_  D  <->  B  C_  { p  e.  ( EE `  N
)  |  ( U 
Btwn  <. Z ,  p >.  \/  p  Btwn  <. Z ,  U >. ) } )
15 ssrab 3337 . . 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 935    = wceq 1647    e. wcel 1715    =/= wne 2529   A.wral 2628   {crab 2632    C_ wss 3238   <.cop 3732   class class class wbr 4125   ` cfv 5358   NNcn 9893   EEcee 25258    Btwn cbtwn 25259
This theorem is referenced by:  axcontlem9  25342  axcontlem10  25343
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ral 2633  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-br 4126
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