Theorem bj-0nelsngl 36156

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 8470). (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 3477	. . . . . 6 ⊢ 𝑥 ∈ V
2	1	snnz 4780	. . . . 5 ⊢ {𝑥} ≠ ∅
3	2	nesymi 2997	. . . 4 ⊢ ¬ ∅ = {𝑥}
4	3	nex 1801	. . 3 ⊢ ¬ ∃𝑥∅ = {𝑥}
5		bj-elsngl 36153	. . . 4 ⊢ (∅ ∈ sngl 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∅ = {𝑥})
6		rexex 3075	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∅ = {𝑥} → ∃𝑥∅ = {𝑥})
7	5, 6	sylbi 216	. . 3 ⊢ (∅ ∈ sngl 𝐴 → ∃𝑥∅ = {𝑥})
8	4, 7	mto 196	. 2 ⊢ ¬ ∅ ∈ sngl 𝐴
9	8	nelir 3048	1 ⊢ ∅ ∉ sngl 𝐴

Syntax hints:   = wceq 1540  ∃wex 1780   ∈ wcel 2105   ∉ wnel 3045  ∃wrex 3069  ∅c0 4322  {csn 4628  sngl bj-csngl 36150

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-nel 3046  df-rex 3070  df-v 3475  df-dif 3951  df-un 3953  df-nul 4323  df-sn 4629  df-pr 4631  df-bj-sngl 36151

This theorem is referenced by: (None)