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Theorem bj-0nelsngl 31635
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 7200). (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 3034 . . . . . 6  |-  x  e. 
_V
21snnz 4081 . . . . 5  |-  { x }  =/=  (/)
32nesymi 2700 . . . 4  |-  -.  (/)  =  {
x }
43nex 1686 . . 3  |-  -.  E. x (/)  =  { x }
5 bj-elsngl 31632 . . . 4  |-  ( (/)  e. sngl  A  <->  E. x  e.  A  (/)  =  { x }
)
6 rexex 2843 . . . 4  |-  ( E. x  e.  A  (/)  =  { x }  ->  E. x (/)  =  {
x } )
75, 6sylbi 200 . . 3  |-  ( (/)  e. sngl  A  ->  E. x (/)  =  { x }
)
84, 7mto 181 . 2  |-  -.  (/)  e. sngl  A
98nelir 2746 1  |-  (/)  e/ sngl  A
Colors of variables: wff setvar class
Syntax hints:    = wceq 1452   E.wex 1671    e. wcel 1904    e/ wnel 2642   E.wrex 2757   (/)c0 3722   {csn 3959  sngl bj-csngl 31629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-v 3033  df-dif 3393  df-un 3395  df-nul 3723  df-sn 3960  df-pr 3962  df-bj-sngl 31630
This theorem is referenced by: (None)
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