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Theorem bj-0nelsngl 33084
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 7605). (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 3234 . . . . . 6 𝑥 ∈ V
21snnz 4340 . . . . 5 {𝑥} ≠ ∅
32nesymi 2880 . . . 4 ¬ ∅ = {𝑥}
43nex 1771 . . 3 ¬ ∃𝑥∅ = {𝑥}
5 bj-elsngl 33081 . . . 4 (∅ ∈ sngl 𝐴 ↔ ∃𝑥𝐴 ∅ = {𝑥})
6 rexex 3031 . . . 4 (∃𝑥𝐴 ∅ = {𝑥} → ∃𝑥∅ = {𝑥})
75, 6sylbi 207 . . 3 (∅ ∈ sngl 𝐴 → ∃𝑥∅ = {𝑥})
84, 7mto 188 . 2 ¬ ∅ ∈ sngl 𝐴
98nelir 2929 1 ∅ ∉ sngl 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1523  wex 1744  wcel 2030  wnel 2926  wrex 2942  c0 3948  {csn 4210  sngl bj-csngl 33078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-v 3233  df-dif 3610  df-un 3612  df-nul 3949  df-sn 4211  df-pr 4213  df-bj-sngl 33079
This theorem is referenced by: (None)
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