Theorem kur14lem1 31188

Description: Lemma for kur14 31198. (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

Step	Hyp	Ref	Expression
1		kur14lem1.a	. . 3 |- A C_ X
2		sseq1 3626	. . 3 |- ( N = A -> ( N C_ X <-> A C_ X ) )
3	1, 2	mpbiri 248	. 2 |- ( N = A -> N C_ X )
4		difeq2 3722	. . . 4 |- ( N = A -> ( X \ N ) = ( X \ A ) )
5		fveq2 6191	. . . 4 |- ( N = A -> ( K ` N ) = ( K ` A ) )
6	4, 5	preq12d 4276	. . 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 4353	. . . 4 |- ( ( ( X \ A ) e. T /\ ( K ` A ) e. T ) -> { ( X \ A ) , ( K ` A ) } C_ T )
10	7, 8, 9	mp2an 708	. . 3 |- { ( X \ A ) , ( K ` A ) } C_ T
11	6, 10	syl6eqss 3655	. 2 |- ( N = A -> { ( X \ N ) , ( K ` N ) } C_ T )
12	3, 11	jca 554	1 |- ( N = A -> ( N C_ X /\ { ( X \ N ) , ( K ` N ) } C_ T ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   \ cdif 3571   C_ wss 3574   {cpr 4179   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896

This theorem is referenced by:  kur14lem7  31194