Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  axcontlem3 Unicode version

Theorem axcontlem3 25817
Description: Lemma for axcont 25827. 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 1000 . 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 964 . . . . . 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 552 . . . . 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 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 ) )  ->  A. x  e.  A  A. y  e.  B  x  Btwn  <. Z ,  y
>. )
54adantr 452 . . . . 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 4183 . . . . . 6  |-  ( x  =  U  ->  (
x  Btwn  <. Z , 
y >. 
<->  U  Btwn  <. Z , 
y >. ) )
7 opeq2 3953 . . . . . . 7  |-  ( y  =  p  ->  <. Z , 
y >.  =  <. Z ,  p >. )
87breq2d 4192 . . . . . 6  |-  ( y  =  p  ->  ( U  Btwn  <. Z ,  y
>. 
<->  U  Btwn  <. Z ,  p >. ) )
96, 8rspc2v 3026 . . . . 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 58 . . . 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 382 . . 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 2757 . 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 3341 . . 3  |-  ( B 
C_  D  <->  B  C_  { p  e.  ( EE `  N
)  |  ( U 
Btwn  <. Z ,  p >.  \/  p  Btwn  <. Z ,  U >. ) } )
15 ssrab 3389 . . 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 241 . 2  |-  ( B 
C_  D  <->  ( B  C_  ( EE `  N
)  /\  A. p  e.  B  ( U  Btwn  <. Z ,  p >.  \/  p  Btwn  <. Z ,  U >. ) ) )
171, 12, 16sylanbrc 646 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 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2575   A.wral 2674   {crab 2678    C_ wss 3288   <.cop 3785   class class class wbr 4180   ` cfv 5421   NNcn 9964   EEcee 25739    Btwn cbtwn 25740
This theorem is referenced by:  axcontlem9  25823  axcontlem10  25824
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ral 2679  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-br 4181
  Copyright terms: Public domain W3C validator