Theorem bj-0nelsngl 38061

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 8501). (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 3458	. . . . . 6 ⊢ 𝑥 ∈ V
2	1	snnz 4737	. . . . 5 ⊢ {𝑥} ≠ ∅
3	2	nesymi 3016	. . . 4 ⊢ ¬ ∅ = {𝑥}
4	3	nex 1833	. . 3 ⊢ ¬ ∃𝑥∅ = {𝑥}
5		bj-elsngl 38058	. . . 4 ⊢ (∅ ∈ sngl 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∅ = {𝑥})
6		rexex 3096	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∅ = {𝑥} → ∃𝑥∅ = {𝑥})
7	5, 6	sylbi 220	. . 3 ⊢ (∅ ∈ sngl 𝐴 → ∃𝑥∅ = {𝑥})
8	4, 7	mto 200	. 2 ⊢ ¬ ∅ ∈ sngl 𝐴
9	8	nelir 3068	1 ⊢ ∅ ∉ sngl 𝐴

This proof depends on syntax axioms:   = wceq 1570  ∃wex 1812   ∈ wcel 2147   ∉ wnel 3065  ∃wrex 3090  ∅c0 4281  {csn 4588  sngl bj-csngl 38055

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-nel 3066  df-rex 3091  df-v 3456  df-dif 4050  df-un 4154  df-nul 4282  df-sn 4589  df-pr 4591  df-bj-sngl 38056

This theorem is used by: (None)