Theorem bj-0nelsngl 37702

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 8458). (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 3457	. . . . . 6 ⊢ 𝑥 ∈ V
2	1	snnz 4740	. . . . 5 ⊢ {𝑥} ≠ ∅
3	2	nesymi 3014	. . . 4 ⊢ ¬ ∅ = {𝑥}
4	3	nex 1833	. . 3 ⊢ ¬ ∃𝑥∅ = {𝑥}
5		bj-elsngl 37699	. . . 4 ⊢ (∅ ∈ sngl 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∅ = {𝑥})
6		rexex 3094	. . . 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 3066	1 ⊢ ∅ ∉ sngl 𝐴

This proof depends on syntax axioms:   = wceq 1570  ∃wex 1812   ∈ wcel 2145   ∉ wnel 3063  ∃wrex 3088  ∅c0 4282  {csn 4587  sngl bj-csngl 37696

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-pr 5402

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 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-nel 3064  df-rex 3089  df-v 3455  df-dif 3905  df-un 3907  df-nul 4283  df-sn 4588  df-pr 4590  df-bj-sngl 37697

This theorem is used by: (None)