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Theorem bj-0nelsngl 31565
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 7182). (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 3048 . . . . . 6  |-  x  e. 
_V
21snnz 4090 . . . . 5  |-  { x }  =/=  (/)
32nesymi 2681 . . . 4  |-  -.  (/)  =  {
x }
43nex 1678 . . 3  |-  -.  E. x (/)  =  { x }
5 bj-elsngl 31562 . . . 4  |-  ( (/)  e. sngl  A  <->  E. x  e.  A  (/)  =  { x }
)
6 rexex 2844 . . . 4  |-  ( E. x  e.  A  (/)  =  { x }  ->  E. x (/)  =  {
x } )
75, 6sylbi 199 . . 3  |-  ( (/)  e. sngl  A  ->  E. x (/)  =  { x }
)
84, 7mto 180 . 2  |-  -.  (/)  e. sngl  A
98nelir 2727 1  |-  (/)  e/ sngl  A
Colors of variables: wff setvar class
Syntax hints:    = wceq 1444   E.wex 1663    e. wcel 1887    e/ wnel 2623   E.wrex 2738   (/)c0 3731   {csn 3968  sngl bj-csngl 31559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-v 3047  df-dif 3407  df-un 3409  df-nul 3732  df-sn 3969  df-pr 3971  df-bj-sngl 31560
This theorem is referenced by: (None)
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