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Theorem enp1ilem 7706
Description: Lemma for uses of enp1i 7707. (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 3587 . . 3  |-  ( ( A  \  { x } )  =  S  ->  ( ( A 
\  { x }
)  u.  { x } )  =  ( S  u.  { x } ) )
2 undif1 3844 . . 3  |-  ( ( A  \  { x } )  u.  {
x } )  =  ( A  u.  {
x } )
3 uncom 3584 . . . 4  |-  ( S  u.  { x }
)  =  ( { x }  u.  S
)
4 enp1ilem.1 . . . 4  |-  T  =  ( { x }  u.  S )
53, 4eqtr4i 2432 . . 3  |-  ( S  u.  { x }
)  =  T
61, 2, 53eqtr3g 2464 . 2  |-  ( ( A  \  { x } )  =  S  ->  ( A  u.  { x } )  =  T )
7 snssi 4113 . . . 4  |-  ( x  e.  A  ->  { x }  C_  A )
8 ssequn2 3613 . . . 4  |-  ( { x }  C_  A  <->  ( A  u.  { x } )  =  A )
97, 8sylib 196 . . 3  |-  ( x  e.  A  ->  ( A  u.  { x } )  =  A )
109eqeq1d 2402 . 2  |-  ( x  e.  A  ->  (
( A  u.  {
x } )  =  T  <->  A  =  T
) )
116, 10syl5ib 219 1  |-  ( x  e.  A  ->  (
( A  \  {
x } )  =  S  ->  A  =  T ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1403    e. wcel 1840    \ cdif 3408    u. cun 3409    C_ wss 3411   {csn 3969
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rab 2760  df-v 3058  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-sn 3970
This theorem is referenced by:  en2  7708  en3  7709  en4  7710
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