Theorem bj-0nelsngl 36959

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 8434). (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.

Step	Hyp	Ref	Expression
1		vex 3451	. . . . . 6 ⊢ 𝑥 ∈ V
2	1	snnz 4740	. . . . 5 ⊢ {𝑥} ≠ ∅
3	2	nesymi 2982	. . . 4 ⊢ ¬ ∅ = {𝑥}
4	3	nex 1800	. . 3 ⊢ ¬ ∃𝑥∅ = {𝑥}
5		bj-elsngl 36956	. . . 4 ⊢ (∅ ∈ sngl 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∅ = {𝑥})
6		rexex 3059	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∅ = {𝑥} → ∃𝑥∅ = {𝑥})
7	5, 6	sylbi 217	. . 3 ⊢ (∅ ∈ sngl 𝐴 → ∃𝑥∅ = {𝑥})
8	4, 7	mto 197	. 2 ⊢ ¬ ∅ ∈ sngl 𝐴
9	8	nelir 3032	1 ⊢ ∅ ∉ sngl 𝐴

Syntax hints:   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ∉ wnel 3029  ∃wrex 3053  ∅c0 4296  {csn 4589  sngl bj-csngl 36953

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 2008  ax-8 2111  ax-9 2119  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-nel 3030  df-rex 3054  df-v 3449  df-dif 3917  df-un 3919  df-nul 4297  df-sn 4590  df-pr 4592  df-bj-sngl 36954

This theorem is referenced by: (None)