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Theorem enp1ilem 7092
Description: Lemma for uses of enp1i 7093. (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 3322 . . 3  |-  ( ( A  \  { x } )  =  S  ->  ( ( A 
\  { x }
)  u.  { x } )  =  ( S  u.  { x } ) )
2 undif1 3529 . . 3  |-  ( ( A  \  { x } )  u.  {
x } )  =  ( A  u.  {
x } )
3 uncom 3319 . . . 4  |-  ( S  u.  { x }
)  =  ( { x }  u.  S
)
4 enp1ilem.1 . . . 4  |-  T  =  ( { x }  u.  S )
53, 4eqtr4i 2306 . . 3  |-  ( S  u.  { x }
)  =  T
61, 2, 53eqtr3g 2338 . 2  |-  ( ( A  \  { x } )  =  S  ->  ( A  u.  { x } )  =  T )
7 snssi 3759 . . . 4  |-  ( x  e.  A  ->  { x }  C_  A )
8 ssequn2 3348 . . . 4  |-  ( { x }  C_  A  <->  ( A  u.  { x } )  =  A )
97, 8sylib 188 . . 3  |-  ( x  e.  A  ->  ( A  u.  { x } )  =  A )
109eqeq1d 2291 . 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 1623    e. wcel 1684    \ cdif 3149    u. cun 3150    C_ wss 3152   {csn 3640
This theorem is referenced by:  en2  7094  en3  7095  en4  7096
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-sn 3646
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