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Theorem kur14lem1 23109
Description: Lemma for kur14 23119. (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 3174 . . 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 3745 . . . 4  |-  ( ( ( X  \  A
)  e.  T  /\  ( K `  A )  e.  T )  ->  { ( X  \  A ) ,  ( K `  A ) }  C_  T )
74, 5, 6mp2an 656 . . 3  |-  { ( X  \  A ) ,  ( K `  A ) }  C_  T
8 difeq2 3263 . . . . 5  |-  ( N  =  A  ->  ( X  \  N )  =  ( X  \  A
) )
9 fveq2 5458 . . . . 5  |-  ( N  =  A  ->  ( K `  N )  =  ( K `  A ) )
108, 9preq12d 3688 . . . 4  |-  ( N  =  A  ->  { ( X  \  N ) ,  ( K `  N ) }  =  { ( X  \  A ) ,  ( K `  A ) } )
1110sseq1d 3180 . . 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 1619    e. wcel 1621    \ cdif 3124    C_ wss 3127   {cpr 3615   ` cfv 4673
This theorem is referenced by:  kur14lem7  23115
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-xp 4675  df-cnv 4677  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fv 4689
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