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Theorem eqsn 4029
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 3403 . . 3  |-  ( A  =  { B }  ->  A  C_  { B } )
2 df-ne 2603 . . . . 5  |-  ( A  =/=  (/)  <->  -.  A  =  (/) )
3 sssn 4026 . . . . . . 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 3341 . . 3  |-  ( A 
C_  { B }  <->  A. x  e.  A  x  e.  { B }
)
10 elsn 3886 . . . 4  |-  ( x  e.  { B }  <->  x  =  B )
1110ralbii 2734 . . 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 1369    e. wcel 1756    =/= wne 2601   A.wral 2710    C_ wss 3323   (/)c0 3632   {csn 3872
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-v 2969  df-dif 3326  df-in 3330  df-ss 3337  df-nul 3633  df-sn 3873
This theorem is referenced by:  zornn0g  8666  hashgt12el  12165  hashgt12el2  12166  hashge2el2dif  12176  lssne0  17009  qtopeu  19264  rngoueqz  23868  lmod0rng  30730
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