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Theorem bj-0nelsngl 33619
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 7130). (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 3116 . . . . . 6  |-  x  e. 
_V
21snnz 4145 . . . . 5  |-  { x }  =/=  (/)
32nesymi 2740 . . . 4  |-  -.  (/)  =  {
x }
43nex 1610 . . 3  |-  -.  E. x (/)  =  { x }
5 bj-elsngl 33616 . . . 4  |-  ( (/)  e. sngl  A  <->  E. x  e.  A  (/)  =  { x }
)
6 rexex 2921 . . . 4  |-  ( E. x  e.  A  (/)  =  { x }  ->  E. x (/)  =  {
x } )
75, 6sylbi 195 . . 3  |-  ( (/)  e. sngl  A  ->  E. x (/)  =  { x }
)
84, 7mto 176 . 2  |-  -.  (/)  e. sngl  A
98nelir 2803 1  |-  (/)  e/ sngl  A
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379   E.wex 1596    e. wcel 1767    e/ wnel 2663   E.wrex 2815   (/)c0 3785   {csn 4027  sngl bj-csngl 33613
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-v 3115  df-dif 3479  df-un 3481  df-nul 3786  df-sn 4028  df-pr 4030  df-bj-sngl 33614
This theorem is referenced by: (None)
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