Theorem bj-0nelsngl 37461

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 8439). (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 3460	. . . . . 6 ⊢ 𝑥 ∈ V
2	1	snnz 4737	. . . . 5 ⊢ {𝑥} ≠ ∅
3	2	nesymi 3016	. . . 4 ⊢ ¬ ∅ = {𝑥}
4	3	nex 1822	. . 3 ⊢ ¬ ∃𝑥∅ = {𝑥}
5		bj-elsngl 37458	. . . 4 ⊢ (∅ ∈ sngl 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∅ = {𝑥})
6		rexex 3094	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∅ = {𝑥} → ∃𝑥∅ = {𝑥})
7	5, 6	sylbi 219	. . 3 ⊢ (∅ ∈ sngl 𝐴 → ∃𝑥∅ = {𝑥})
8	4, 7	mto 199	. 2 ⊢ ¬ ∅ ∈ sngl 𝐴
9	8	nelir 3066	1 ⊢ ∅ ∉ sngl 𝐴

Syntax hints:   = wceq 1562  ∃wex 1801   ∈ wcel 2144   ∉ wnel 3063  ∃wrex 3088  ∅c0 4287  {csn 4584  sngl bj-csngl 37455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ne 2960  df-nel 3064  df-rex 3089  df-v 3458  df-dif 3909  df-un 3911  df-nul 4288  df-sn 4585  df-pr 4587  df-bj-sngl 37456

This theorem is referenced by: (None)