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Theorem eqsnm 3599
Description: Two ways to express that an inhabited set equals a singleton. (Contributed by Jim Kingdon, 11-Aug-2018.)
Assertion
Ref Expression
eqsnm  |-  ( E. x  x  e.  A  ->  ( A  =  { B }  <->  A. x  e.  A  x  =  B )
)
Distinct variable groups:    x, A    x, B

Proof of Theorem eqsnm
StepHypRef Expression
1 dfss3 3015 . . 3  |-  ( A 
C_  { B }  <->  A. x  e.  A  x  e.  { B }
)
2 velsn 3463 . . . 4  |-  ( x  e.  { B }  <->  x  =  B )
32ralbii 2384 . . 3  |-  ( A. x  e.  A  x  e.  { B }  <->  A. x  e.  A  x  =  B )
41, 3bitri 182 . 2  |-  ( A 
C_  { B }  <->  A. x  e.  A  x  =  B )
5 sssnm 3598 . 2  |-  ( E. x  x  e.  A  ->  ( A  C_  { B } 
<->  A  =  { B } ) )
64, 5syl5rbbr 193 1  |-  ( E. x  x  e.  A  ->  ( A  =  { B }  <->  A. x  e.  A  x  =  B )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1289   E.wex 1426    e. wcel 1438   A.wral 2359    C_ wss 2999   {csn 3446
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-in 3005  df-ss 3012  df-sn 3452
This theorem is referenced by:  nninfall  11855
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