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Theorem kur14lem1 23141
Description: Lemma for kur14 23151. (Contributed by Mario Carneiro, 17-Feb-2015.)
Hypotheses
Ref Expression
kur14lem1.a  |-  A  C_  X
kur14lem1.c  |-  ( X 
\  A )  e.  T
kur14lem1.k  |-  ( K `
 A )  e.  T
Assertion
Ref Expression
kur14lem1  |-  ( N  =  A  ->  ( N  C_  X  /\  {
( X  \  N
) ,  ( K `
 N ) } 
C_  T ) )

Proof of Theorem kur14lem1
StepHypRef Expression
1 kur14lem1.a . . 3  |-  A  C_  X
2 sseq1 3200 . . 3  |-  ( N  =  A  ->  ( N  C_  X  <->  A  C_  X
) )
31, 2mpbiri 226 . 2  |-  ( N  =  A  ->  N  C_  X )
4 kur14lem1.c . . . 4  |-  ( X 
\  A )  e.  T
5 kur14lem1.k . . . 4  |-  ( K `
 A )  e.  T
6 prssi 3772 . . . 4  |-  ( ( ( X  \  A
)  e.  T  /\  ( K `  A )  e.  T )  ->  { ( X  \  A ) ,  ( K `  A ) }  C_  T )
74, 5, 6mp2an 655 . . 3  |-  { ( X  \  A ) ,  ( K `  A ) }  C_  T
8 difeq2 3289 . . . . 5  |-  ( N  =  A  ->  ( X  \  N )  =  ( X  \  A
) )
9 fveq2 5485 . . . . 5  |-  ( N  =  A  ->  ( K `  N )  =  ( K `  A ) )
108, 9preq12d 3715 . . . 4  |-  ( N  =  A  ->  { ( X  \  N ) ,  ( K `  N ) }  =  { ( X  \  A ) ,  ( K `  A ) } )
1110sseq1d 3206 . . 3  |-  ( N  =  A  ->  ( { ( X  \  N ) ,  ( K `  N ) }  C_  T  <->  { ( X  \  A ) ,  ( K `  A
) }  C_  T
) )
127, 11mpbiri 226 . 2  |-  ( N  =  A  ->  { ( X  \  N ) ,  ( K `  N ) }  C_  T )
133, 12jca 520 1  |-  ( N  =  A  ->  ( N  C_  X  /\  {
( X  \  N
) ,  ( K `
 N ) } 
C_  T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1624    e. wcel 1685    \ cdif 3150    C_ wss 3153   {cpr 3642   ` cfv 5221
This theorem is referenced by:  kur14lem7  23147
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-xp 4694  df-cnv 4696  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fv 5229
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