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Theorem enp1ilem 7334
Description: Lemma for uses of enp1i 7335. (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 3486 . . 3  |-  ( ( A  \  { x } )  =  S  ->  ( ( A 
\  { x }
)  u.  { x } )  =  ( S  u.  { x } ) )
2 undif1 3695 . . 3  |-  ( ( A  \  { x } )  u.  {
x } )  =  ( A  u.  {
x } )
3 uncom 3483 . . . 4  |-  ( S  u.  { x }
)  =  ( { x }  u.  S
)
4 enp1ilem.1 . . . 4  |-  T  =  ( { x }  u.  S )
53, 4eqtr4i 2458 . . 3  |-  ( S  u.  { x }
)  =  T
61, 2, 53eqtr3g 2490 . 2  |-  ( ( A  \  { x } )  =  S  ->  ( A  u.  { x } )  =  T )
7 snssi 3934 . . . 4  |-  ( x  e.  A  ->  { x }  C_  A )
8 ssequn2 3512 . . . 4  |-  ( { x }  C_  A  <->  ( A  u.  { x } )  =  A )
97, 8sylib 189 . . 3  |-  ( x  e.  A  ->  ( A  u.  { x } )  =  A )
109eqeq1d 2443 . 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 1652    e. wcel 1725    \ cdif 3309    u. cun 3310    C_ wss 3312   {csn 3806
This theorem is referenced by:  en2  7336  en3  7337  en4  7338
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-sn 3812
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