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Theorem bj-0nelsngl 33481
 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 7826). (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 3417 . . . . . 6 𝑥 ∈ V
21snnz 4528 . . . . 5 {𝑥} ≠ ∅
32nesymi 3056 . . . 4 ¬ ∅ = {𝑥}
43nex 1901 . . 3 ¬ ∃𝑥∅ = {𝑥}
5 bj-elsngl 33478 . . . 4 (∅ ∈ sngl 𝐴 ↔ ∃𝑥𝐴 ∅ = {𝑥})
6 rexex 3210 . . . 4 (∃𝑥𝐴 ∅ = {𝑥} → ∃𝑥∅ = {𝑥})
75, 6sylbi 209 . . 3 (∅ ∈ sngl 𝐴 → ∃𝑥∅ = {𝑥})
84, 7mto 189 . 2 ¬ ∅ ∈ sngl 𝐴
98nelir 3105 1 ∅ ∉ sngl 𝐴
 Colors of variables: wff setvar class Syntax hints:   = wceq 1658  ∃wex 1880   ∈ wcel 2166   ∉ wnel 3102  ∃wrex 3118  ∅c0 4144  {csn 4397  sngl bj-csngl 33475 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-v 3416  df-dif 3801  df-un 3803  df-nul 4145  df-sn 4398  df-pr 4400  df-bj-sngl 33476 This theorem is referenced by: (None)
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