Theorem enp1ilem 8194

Description: Lemma for uses of enp1i 8195. (Contributed by Mario Carneiro, 5-Jan-2016.)

Hypothesis

Ref	Expression
enp1ilem.1	|- T = ( { x } u. S )

Assertion

Ref	Expression
enp1ilem	|- ( x e. A -> ( ( A \ { x } ) = S -> A = T ) )

Proof of Theorem enp1ilem

Step	Hyp	Ref	Expression
1		uneq1 3760	. . 3 |- ( ( A \ { x } ) = S -> ( ( A \ { x } ) u. { x } ) = ( S u. { x } ) )
2		undif1 4043	. . 3 |- ( ( A \ { x } ) u. { x } ) = ( A u. { x } )
3		uncom 3757	. . . 4 |- ( S u. { x } ) = ( { x } u. S )
4		enp1ilem.1	. . . 4 |- T = ( { x } u. S )
5	3, 4	eqtr4i 2647	. . 3 |- ( S u. { x } ) = T
6	1, 2, 5	3eqtr3g 2679	. 2 |- ( ( A \ { x } ) = S -> ( A u. { x } ) = T )
7		snssi 4339	. . . 4 |- ( x e. A -> { x } C_ A )
8		ssequn2 3786	. . . 4 |- ( { x } C_ A <-> ( A u. { x } ) = A )
9	7, 8	sylib 208	. . 3 |- ( x e. A -> ( A u. { x } ) = A )
10	9	eqeq1d 2624	. 2 |- ( x e. A -> ( ( A u. { x } ) = T <-> A = T ) )
11	6, 10	syl5ib 234	1 |- ( x e. A -> ( ( A \ { x } ) = S -> A = T ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   \ cdif 3571   u. cun 3572   C_ wss 3574   {csn 4177

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-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178

This theorem is referenced by:  en2  8196  en3  8197  en4  8198