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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 sngl

Proof of Theorem bj-0nelsngl
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 vex 3048 . . . . . 6
21snnz 4090 . . . . 5
32nesymi 2681 . . . 4
43nex 1678 . . 3
5 bj-elsngl 31562 . . . 4 sngl
6 rexex 2844 . . . 4
75, 6sylbi 199 . . 3 sngl
84, 7mto 180 . 2 sngl
98nelir 2727 1 sngl
 Colors of variables: wff setvar class Syntax hints:   wceq 1444  wex 1663   wcel 1887   wnel 2623  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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