Theorem bj-0nelsngl 37015

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 8385). (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 3440	. . . . . 6 ⊢ 𝑥 ∈ V
2	1	snnz 4726	. . . . 5 ⊢ {𝑥} ≠ ∅
3	2	nesymi 2985	. . . 4 ⊢ ¬ ∅ = {𝑥}
4	3	nex 1801	. . 3 ⊢ ¬ ∃𝑥∅ = {𝑥}
5		bj-elsngl 37012	. . . 4 ⊢ (∅ ∈ sngl 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∅ = {𝑥})
6		rexex 3062	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∅ = {𝑥} → ∃𝑥∅ = {𝑥})
7	5, 6	sylbi 217	. . 3 ⊢ (∅ ∈ sngl 𝐴 → ∃𝑥∅ = {𝑥})
8	4, 7	mto 197	. 2 ⊢ ¬ ∅ ∈ sngl 𝐴
9	8	nelir 3035	1 ⊢ ∅ ∉ sngl 𝐴

Syntax hints:   = wceq 1541  ∃wex 1780   ∈ wcel 2111   ∉ wnel 3032  ∃wrex 3056  ∅c0 4280  {csn 4573  sngl bj-csngl 37009

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 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-nel 3033  df-rex 3057  df-v 3438  df-dif 3900  df-un 3902  df-nul 4281  df-sn 4574  df-pr 4576  df-bj-sngl 37010

This theorem is referenced by: (None)