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Theorem bj-0nelsngl 36972
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 8506). (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 3484 . . . . . 6 𝑥 ∈ V
21snnz 4776 . . . . 5 {𝑥} ≠ ∅
32nesymi 2998 . . . 4 ¬ ∅ = {𝑥}
43nex 1800 . . 3 ¬ ∃𝑥∅ = {𝑥}
5 bj-elsngl 36969 . . . 4 (∅ ∈ sngl 𝐴 ↔ ∃𝑥𝐴 ∅ = {𝑥})
6 rexex 3076 . . . 4 (∃𝑥𝐴 ∅ = {𝑥} → ∃𝑥∅ = {𝑥})
75, 6sylbi 217 . . 3 (∅ ∈ sngl 𝐴 → ∃𝑥∅ = {𝑥})
84, 7mto 197 . 2 ¬ ∅ ∈ sngl 𝐴
98nelir 3049 1 ∅ ∉ sngl 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wex 1779  wcel 2108  wnel 3046  wrex 3070  c0 4333  {csn 4626  sngl bj-csngl 36966
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-nel 3047  df-rex 3071  df-v 3482  df-dif 3954  df-un 3956  df-nul 4334  df-sn 4627  df-pr 4629  df-bj-sngl 36967
This theorem is referenced by: (None)
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