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Theorem eqsn 3791
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 3243 . . 3  |-  ( A  =  { B }  ->  A  C_  { B } )
2 df-ne 2461 . . . . 5  |-  ( A  =/=  (/)  <->  -.  A  =  (/) )
3 sssn 3788 . . . . . . 7  |-  ( A 
C_  { B }  <->  ( A  =  (/)  \/  A  =  { B } ) )
43biimpi 186 . . . . . 6  |-  ( A 
C_  { B }  ->  ( A  =  (/)  \/  A  =  { B } ) )
54ord 366 . . . . 5  |-  ( A 
C_  { B }  ->  ( -.  A  =  (/)  ->  A  =  { B } ) )
62, 5syl5bi 208 . . . 4  |-  ( A 
C_  { B }  ->  ( A  =/=  (/)  ->  A  =  { B } ) )
76com12 27 . . 3  |-  ( A  =/=  (/)  ->  ( A  C_ 
{ B }  ->  A  =  { B }
) )
81, 7impbid2 195 . 2  |-  ( A  =/=  (/)  ->  ( A  =  { B }  <->  A  C_  { B } ) )
9 dfss3 3183 . . 3  |-  ( A 
C_  { B }  <->  A. x  e.  A  x  e.  { B }
)
10 elsn 3668 . . . 4  |-  ( x  e.  { B }  <->  x  =  B )
1110ralbii 2580 . . 3  |-  ( A. x  e.  A  x  e.  { B }  <->  A. x  e.  A  x  =  B )
129, 11bitri 240 . 2  |-  ( A 
C_  { B }  <->  A. x  e.  A  x  =  B )
138, 12syl6bb 252 1  |-  ( A  =/=  (/)  ->  ( A  =  { B }  <->  A. x  e.  A  x  =  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556    C_ wss 3165   (/)c0 3468   {csn 3653
This theorem is referenced by:  zornn0g  8148  lssne0  15724  qtopeu  17423  rngoueqz  21113  hashgt12el  28218  hashgt12el2  28219
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-v 2803  df-dif 3168  df-in 3172  df-ss 3179  df-nul 3469  df-sn 3659
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