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Theorem bj-0nelsngl 36450
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 8486). (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 3475 . . . . . 6 𝑥 ∈ V
21snnz 4781 . . . . 5 {𝑥} ≠ ∅
32nesymi 2995 . . . 4 ¬ ∅ = {𝑥}
43nex 1795 . . 3 ¬ ∃𝑥∅ = {𝑥}
5 bj-elsngl 36447 . . . 4 (∅ ∈ sngl 𝐴 ↔ ∃𝑥𝐴 ∅ = {𝑥})
6 rexex 3073 . . . 4 (∃𝑥𝐴 ∅ = {𝑥} → ∃𝑥∅ = {𝑥})
75, 6sylbi 216 . . 3 (∅ ∈ sngl 𝐴 → ∃𝑥∅ = {𝑥})
84, 7mto 196 . 2 ¬ ∅ ∈ sngl 𝐴
98nelir 3046 1 ∅ ∉ sngl 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1534  wex 1774  wcel 2099  wnel 3043  wrex 3067  c0 4323  {csn 4629  sngl bj-csngl 36444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-nel 3044  df-rex 3068  df-v 3473  df-dif 3950  df-un 3952  df-nul 4324  df-sn 4630  df-pr 4632  df-bj-sngl 36445
This theorem is referenced by: (None)
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