Theorem bj-0nelsngl 36954

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 8505). (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 3482	. . . . . 6 ⊢ 𝑥 ∈ V
2	1	snnz 4781	. . . . 5 ⊢ {𝑥} ≠ ∅
3	2	nesymi 2996	. . . 4 ⊢ ¬ ∅ = {𝑥}
4	3	nex 1797	. . 3 ⊢ ¬ ∃𝑥∅ = {𝑥}
5		bj-elsngl 36951	. . . 4 ⊢ (∅ ∈ sngl 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∅ = {𝑥})
6		rexex 3074	. . . 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 3047	1 ⊢ ∅ ∉ sngl 𝐴

Syntax hints:   = wceq 1537  ∃wex 1776   ∈ wcel 2106   ∉ wnel 3044  ∃wrex 3068  ∅c0 4339  {csn 4631  sngl bj-csngl 36948

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-nel 3045  df-rex 3069  df-v 3480  df-dif 3966  df-un 3968  df-nul 4340  df-sn 4632  df-pr 4634  df-bj-sngl 36949

This theorem is referenced by: (None)