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Theorem eqsn 4135
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 3509 . . 3  |-  ( A  =  { B }  ->  A  C_  { B } )
2 df-ne 2646 . . . . 5  |-  ( A  =/=  (/)  <->  -.  A  =  (/) )
3 sssn 4132 . . . . . . 7  |-  ( A 
C_  { B }  <->  ( A  =  (/)  \/  A  =  { B } ) )
43biimpi 194 . . . . . 6  |-  ( A 
C_  { B }  ->  ( A  =  (/)  \/  A  =  { B } ) )
54ord 377 . . . . 5  |-  ( A 
C_  { B }  ->  ( -.  A  =  (/)  ->  A  =  { B } ) )
62, 5syl5bi 217 . . . 4  |-  ( A 
C_  { B }  ->  ( A  =/=  (/)  ->  A  =  { B } ) )
76com12 31 . . 3  |-  ( A  =/=  (/)  ->  ( A  C_ 
{ B }  ->  A  =  { B }
) )
81, 7impbid2 204 . 2  |-  ( A  =/=  (/)  ->  ( A  =  { B }  <->  A  C_  { B } ) )
9 dfss3 3447 . . 3  |-  ( A 
C_  { B }  <->  A. x  e.  A  x  e.  { B }
)
10 elsn 3992 . . . 4  |-  ( x  e.  { B }  <->  x  =  B )
1110ralbii 2834 . . 3  |-  ( A. x  e.  A  x  e.  { B }  <->  A. x  e.  A  x  =  B )
129, 11bitri 249 . 2  |-  ( A 
C_  { B }  <->  A. x  e.  A  x  =  B )
138, 12syl6bb 261 1  |-  ( A  =/=  (/)  ->  ( A  =  { B }  <->  A. x  e.  A  x  =  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    = wceq 1370    e. wcel 1758    =/= wne 2644   A.wral 2795    C_ wss 3429   (/)c0 3738   {csn 3978
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-v 3073  df-dif 3432  df-in 3436  df-ss 3443  df-nul 3739  df-sn 3979
This theorem is referenced by:  zornn0g  8778  hashgt12el  12284  hashgt12el2  12285  hashge2el2dif  12295  lssne0  17147  qtopeu  19414  rngoueqz  24062  lmod0rng  30920
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