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Theorem enp1ilem 7280
Description: Lemma for uses of enp1i 7281. (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
StepHypRef Expression
1 uneq1 3439 . . 3  |-  ( ( A  \  { x } )  =  S  ->  ( ( A 
\  { x }
)  u.  { x } )  =  ( S  u.  { x } ) )
2 undif1 3648 . . 3  |-  ( ( A  \  { x } )  u.  {
x } )  =  ( A  u.  {
x } )
3 uncom 3436 . . . 4  |-  ( S  u.  { x }
)  =  ( { x }  u.  S
)
4 enp1ilem.1 . . . 4  |-  T  =  ( { x }  u.  S )
53, 4eqtr4i 2412 . . 3  |-  ( S  u.  { x }
)  =  T
61, 2, 53eqtr3g 2444 . 2  |-  ( ( A  \  { x } )  =  S  ->  ( A  u.  { x } )  =  T )
7 snssi 3887 . . . 4  |-  ( x  e.  A  ->  { x }  C_  A )
8 ssequn2 3465 . . . 4  |-  ( { x }  C_  A  <->  ( A  u.  { x } )  =  A )
97, 8sylib 189 . . 3  |-  ( x  e.  A  ->  ( A  u.  { x } )  =  A )
109eqeq1d 2397 . 2  |-  ( x  e.  A  ->  (
( A  u.  {
x } )  =  T  <->  A  =  T
) )
116, 10syl5ib 211 1  |-  ( x  e.  A  ->  (
( A  \  {
x } )  =  S  ->  A  =  T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717    \ cdif 3262    u. cun 3263    C_ wss 3265   {csn 3759
This theorem is referenced by:  en2  7282  en3  7283  en4  7284
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-sn 3765
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