Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  kur14lem1 Structured version   Unicode version

Theorem kur14lem1 28276
Description: Lemma for kur14 28286. (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 3518 . . 3  |-  ( N  =  A  ->  ( N  C_  X  <->  A  C_  X
) )
31, 2mpbiri 233 . 2  |-  ( N  =  A  ->  N  C_  X )
4 difeq2 3609 . . . 4  |-  ( N  =  A  ->  ( X  \  N )  =  ( X  \  A
) )
5 fveq2 5857 . . . 4  |-  ( N  =  A  ->  ( K `  N )  =  ( K `  A ) )
64, 5preq12d 4107 . . 3  |-  ( N  =  A  ->  { ( X  \  N ) ,  ( K `  N ) }  =  { ( X  \  A ) ,  ( K `  A ) } )
7 kur14lem1.c . . . 4  |-  ( X 
\  A )  e.  T
8 kur14lem1.k . . . 4  |-  ( K `
 A )  e.  T
9 prssi 4176 . . . 4  |-  ( ( ( X  \  A
)  e.  T  /\  ( K `  A )  e.  T )  ->  { ( X  \  A ) ,  ( K `  A ) }  C_  T )
107, 8, 9mp2an 672 . . 3  |-  { ( X  \  A ) ,  ( K `  A ) }  C_  T
116, 10syl6eqss 3547 . 2  |-  ( N  =  A  ->  { ( X  \  N ) ,  ( K `  N ) }  C_  T )
123, 11jca 532 1  |-  ( N  =  A  ->  ( N  C_  X  /\  {
( X  \  N
) ,  ( K `
 N ) } 
C_  T ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762    \ cdif 3466    C_ wss 3469   {cpr 4022   ` cfv 5579
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-iota 5542  df-fv 5587
This theorem is referenced by:  kur14lem7  28282
  Copyright terms: Public domain W3C validator