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Theorem bj-0nelsngl 31314
Description: The empty set is not a member of a singletonization (neither is any nonsingleton, in particular any von Neuman ordinal except possibly df-1o 7190). (Contributed by BJ, 6-Oct-2018.)
Assertion
Ref Expression
bj-0nelsngl  |-  (/)  e/ sngl  A

Proof of Theorem bj-0nelsngl
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 vex 3090 . . . . . 6  |-  x  e. 
_V
21snnz 4121 . . . . 5  |-  { x }  =/=  (/)
32nesymi 2704 . . . 4  |-  -.  (/)  =  {
x }
43nex 1674 . . 3  |-  -.  E. x (/)  =  { x }
5 bj-elsngl 31311 . . . 4  |-  ( (/)  e. sngl  A  <->  E. x  e.  A  (/)  =  { x }
)
6 rexex 2889 . . . 4  |-  ( E. x  e.  A  (/)  =  { x }  ->  E. x (/)  =  {
x } )
75, 6sylbi 198 . . 3  |-  ( (/)  e. sngl  A  ->  E. x (/)  =  { x }
)
84, 7mto 179 . 2  |-  -.  (/)  e. sngl  A
98nelir 2768 1  |-  (/)  e/ sngl  A
Colors of variables: wff setvar class
Syntax hints:    = wceq 1437   E.wex 1659    e. wcel 1870    e/ wnel 2626   E.wrex 2783   (/)c0 3767   {csn 4002  sngl bj-csngl 31308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-v 3089  df-dif 3445  df-un 3447  df-nul 3768  df-sn 4003  df-pr 4005  df-bj-sngl 31309
This theorem is referenced by: (None)
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