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Theorem kur14lem1 23737
Description: Lemma for kur14 23747. (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 3199 . . 3  |-  ( N  =  A  ->  ( N  C_  X  <->  A  C_  X
) )
31, 2mpbiri 224 . 2  |-  ( N  =  A  ->  N  C_  X )
4 kur14lem1.c . . . 4  |-  ( X 
\  A )  e.  T
5 kur14lem1.k . . . 4  |-  ( K `
 A )  e.  T
6 prssi 3771 . . . 4  |-  ( ( ( X  \  A
)  e.  T  /\  ( K `  A )  e.  T )  ->  { ( X  \  A ) ,  ( K `  A ) }  C_  T )
74, 5, 6mp2an 653 . . 3  |-  { ( X  \  A ) ,  ( K `  A ) }  C_  T
8 difeq2 3288 . . . . 5  |-  ( N  =  A  ->  ( X  \  N )  =  ( X  \  A
) )
9 fveq2 5525 . . . . 5  |-  ( N  =  A  ->  ( K `  N )  =  ( K `  A ) )
108, 9preq12d 3714 . . . 4  |-  ( N  =  A  ->  { ( X  \  N ) ,  ( K `  N ) }  =  { ( X  \  A ) ,  ( K `  A ) } )
1110sseq1d 3205 . . 3  |-  ( N  =  A  ->  ( { ( X  \  N ) ,  ( K `  N ) }  C_  T  <->  { ( X  \  A ) ,  ( K `  A
) }  C_  T
) )
127, 11mpbiri 224 . 2  |-  ( N  =  A  ->  { ( X  \  N ) ,  ( K `  N ) }  C_  T )
133, 12jca 518 1  |-  ( N  =  A  ->  ( N  C_  X  /\  {
( X  \  N
) ,  ( K `
 N ) } 
C_  T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    \ cdif 3149    C_ wss 3152   {cpr 3641   ` cfv 5255
This theorem is referenced by:  kur14lem7  23743
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263
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