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

Theorem kur14lem1 29490
Description: Lemma for kur14 29500. (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 3462 . . 3  |-  ( N  =  A  ->  ( N  C_  X  <->  A  C_  X
) )
31, 2mpbiri 233 . 2  |-  ( N  =  A  ->  N  C_  X )
4 difeq2 3554 . . . 4  |-  ( N  =  A  ->  ( X  \  N )  =  ( X  \  A
) )
5 fveq2 5848 . . . 4  |-  ( N  =  A  ->  ( K `  N )  =  ( K `  A ) )
64, 5preq12d 4058 . . 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 4127 . . . 4  |-  ( ( ( X  \  A
)  e.  T  /\  ( K `  A )  e.  T )  ->  { ( X  \  A ) ,  ( K `  A ) }  C_  T )
107, 8, 9mp2an 670 . . 3  |-  { ( X  \  A ) ,  ( K `  A ) }  C_  T
116, 10syl6eqss 3491 . 2  |-  ( N  =  A  ->  { ( X  \  N ) ,  ( K `  N ) }  C_  T )
123, 11jca 530 1  |-  ( N  =  A  ->  ( N  C_  X  /\  {
( X  \  N
) ,  ( K `
 N ) } 
C_  T ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842    \ cdif 3410    C_ wss 3413   {cpr 3973   ` cfv 5568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-iota 5532  df-fv 5576
This theorem is referenced by:  kur14lem7  29496
  Copyright terms: Public domain W3C validator