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Theorem bj-0nelsngl 33318
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 7764). (Contributed by BJ, 6-Oct-2018.)
Assertion
Ref Expression
bj-0nelsngl ∅ ∉ sngl 𝐴

Proof of Theorem bj-0nelsngl
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 vex 3353 . . . . . 6 𝑥 ∈ V
21snnz 4463 . . . . 5 {𝑥} ≠ ∅
32nesymi 2994 . . . 4 ¬ ∅ = {𝑥}
43nex 1895 . . 3 ¬ ∃𝑥∅ = {𝑥}
5 bj-elsngl 33315 . . . 4 (∅ ∈ sngl 𝐴 ↔ ∃𝑥𝐴 ∅ = {𝑥})
6 rexex 3148 . . . 4 (∃𝑥𝐴 ∅ = {𝑥} → ∃𝑥∅ = {𝑥})
75, 6sylbi 208 . . 3 (∅ ∈ sngl 𝐴 → ∃𝑥∅ = {𝑥})
84, 7mto 188 . 2 ¬ ∅ ∈ sngl 𝐴
98nelir 3043 1 ∅ ∉ sngl 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1652  wex 1874  wcel 2155  wnel 3040  wrex 3056  c0 4079  {csn 4334  sngl bj-csngl 33312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-v 3352  df-dif 3735  df-un 3737  df-nul 4080  df-sn 4335  df-pr 4337  df-bj-sngl 33313
This theorem is referenced by: (None)
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