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Theorem enp1ilem 7108
Description: Lemma for uses of enp1i 7109. (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 3335 . . 3  |-  ( ( A  \  { x } )  =  S  ->  ( ( A 
\  { x }
)  u.  { x } )  =  ( S  u.  { x } ) )
2 undif1 3542 . . 3  |-  ( ( A  \  { x } )  u.  {
x } )  =  ( A  u.  {
x } )
3 uncom 3332 . . . 4  |-  ( S  u.  { x }
)  =  ( { x }  u.  S
)
4 enp1ilem.1 . . . 4  |-  T  =  ( { x }  u.  S )
53, 4eqtr4i 2319 . . 3  |-  ( S  u.  { x }
)  =  T
61, 2, 53eqtr3g 2351 . 2  |-  ( ( A  \  { x } )  =  S  ->  ( A  u.  { x } )  =  T )
7 snssi 3775 . . . 4  |-  ( x  e.  A  ->  { x }  C_  A )
8 ssequn2 3361 . . . 4  |-  ( { x }  C_  A  <->  ( A  u.  { x } )  =  A )
97, 8sylib 188 . . 3  |-  ( x  e.  A  ->  ( A  u.  { x } )  =  A )
109eqeq1d 2304 . 2  |-  ( x  e.  A  ->  (
( A  u.  {
x } )  =  T  <->  A  =  T
) )
116, 10syl5ib 210 1  |-  ( x  e.  A  ->  (
( A  \  {
x } )  =  S  ->  A  =  T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696    \ cdif 3162    u. cun 3163    C_ wss 3165   {csn 3653
This theorem is referenced by:  en2  7110  en3  7111  en4  7112
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-sn 3659
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