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Theorem bj-0nelsngl 34300
 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 8080). (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 3476 . . . . . 6 𝑥 ∈ V
21snnz 4687 . . . . 5 {𝑥} ≠ ∅
32nesymi 3063 . . . 4 ¬ ∅ = {𝑥}
43nex 1801 . . 3 ¬ ∃𝑥∅ = {𝑥}
5 bj-elsngl 34297 . . . 4 (∅ ∈ sngl 𝐴 ↔ ∃𝑥𝐴 ∅ = {𝑥})
6 rexex 3227 . . . 4 (∃𝑥𝐴 ∅ = {𝑥} → ∃𝑥∅ = {𝑥})
75, 6sylbi 219 . . 3 (∅ ∈ sngl 𝐴 → ∃𝑥∅ = {𝑥})
84, 7mto 199 . 2 ¬ ∅ ∈ sngl 𝐴
98nelir 3113 1 ∅ ∉ sngl 𝐴
 Colors of variables: wff setvar class Syntax hints:   = wceq 1537  ∃wex 1780   ∈ wcel 2114   ∉ wnel 3110  ∃wrex 3126  ∅c0 4269  {csn 4543  sngl bj-csngl 34294 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pr 5306 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-nel 3111  df-rex 3131  df-v 3475  df-dif 3916  df-un 3918  df-nul 4270  df-sn 4544  df-pr 4546  df-bj-sngl 34295 This theorem is referenced by: (None)
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