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Theorem eqsnm 3773
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 sssnm 3772 . 2  |-  ( E. x  x  e.  A  ->  ( A  C_  { B } 
<->  A  =  { B } ) )
2 dfss3 3160 . . 3  |-  ( A 
C_  { B }  <->  A. x  e.  A  x  e.  { B }
)
3 velsn 3627 . . . 4  |-  ( x  e.  { B }  <->  x  =  B )
43ralbii 2496 . . 3  |-  ( A. x  e.  A  x  e.  { B }  <->  A. x  e.  A  x  =  B )
52, 4bitri 184 . 2  |-  ( A 
C_  { B }  <->  A. x  e.  A  x  =  B )
61, 5bitr3di 195 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 105    = wceq 1364   E.wex 1503    e. wcel 2160   A.wral 2468    C_ wss 3144   {csn 3610
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-in 3150  df-ss 3157  df-sn 3616
This theorem is referenced by:  01eq0ring  13561  nninfall  15245
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