Theorem bj-0nelsngl 37337

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 8399). (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 3437	. . . . . 6 ⊢ 𝑥 ∈ V
2	1	snnz 4710	. . . . 5 ⊢ {𝑥} ≠ ∅
3	2	nesymi 2993	. . . 4 ⊢ ¬ ∅ = {𝑥}
4	3	nex 1808	. . 3 ⊢ ¬ ∃𝑥∅ = {𝑥}
5		bj-elsngl 37334	. . . 4 ⊢ (∅ ∈ sngl 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∅ = {𝑥})
6		rexex 3071	. . . 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 3043	1 ⊢ ∅ ∉ sngl 𝐴

Syntax hints:   = wceq 1548  ∃wex 1787   ∈ wcel 2121   ∉ wnel 3040  ∃wrex 3065  ∅c0 4263  {csn 4557  sngl bj-csngl 37331

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5220  ax-pr 5364

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ne 2937  df-nel 3041  df-rex 3066  df-v 3435  df-dif 3887  df-un 3889  df-nul 4264  df-sn 4558  df-pr 4560  df-bj-sngl 37332

This theorem is referenced by: (None)