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Theorem eqsn 3960
Description: Two ways to express that a nonempty set equals a singleton. (Contributed by NM, 15-Dec-2007.)
Assertion
Ref Expression
eqsn  |-  ( A  =/=  (/)  ->  ( A  =  { B }  <->  A. x  e.  A  x  =  B ) )
Distinct variable groups:    x, A    x, B

Proof of Theorem eqsn
StepHypRef Expression
1 eqimss 3400 . . 3  |-  ( A  =  { B }  ->  A  C_  { B } )
2 df-ne 2601 . . . . 5  |-  ( A  =/=  (/)  <->  -.  A  =  (/) )
3 sssn 3957 . . . . . . 7  |-  ( A 
C_  { B }  <->  ( A  =  (/)  \/  A  =  { B } ) )
43biimpi 187 . . . . . 6  |-  ( A 
C_  { B }  ->  ( A  =  (/)  \/  A  =  { B } ) )
54ord 367 . . . . 5  |-  ( A 
C_  { B }  ->  ( -.  A  =  (/)  ->  A  =  { B } ) )
62, 5syl5bi 209 . . . 4  |-  ( A 
C_  { B }  ->  ( A  =/=  (/)  ->  A  =  { B } ) )
76com12 29 . . 3  |-  ( A  =/=  (/)  ->  ( A  C_ 
{ B }  ->  A  =  { B }
) )
81, 7impbid2 196 . 2  |-  ( A  =/=  (/)  ->  ( A  =  { B }  <->  A  C_  { B } ) )
9 dfss3 3338 . . 3  |-  ( A 
C_  { B }  <->  A. x  e.  A  x  e.  { B }
)
10 elsn 3829 . . . 4  |-  ( x  e.  { B }  <->  x  =  B )
1110ralbii 2729 . . 3  |-  ( A. x  e.  A  x  e.  { B }  <->  A. x  e.  A  x  =  B )
129, 11bitri 241 . 2  |-  ( A 
C_  { B }  <->  A. x  e.  A  x  =  B )
138, 12syl6bb 253 1  |-  ( A  =/=  (/)  ->  ( A  =  { B }  <->  A. x  e.  A  x  =  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705    C_ wss 3320   (/)c0 3628   {csn 3814
This theorem is referenced by:  zornn0g  8385  hashgt12el  11682  hashgt12el2  11683  lssne0  16027  qtopeu  17748  rngoueqz  22018
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 2417
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 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-v 2958  df-dif 3323  df-in 3327  df-ss 3334  df-nul 3629  df-sn 3820
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