Theorem bj-0nelsngl 36450

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 8486). (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 3475	. . . . . 6 ⊢ 𝑥 ∈ V
2	1	snnz 4781	. . . . 5 ⊢ {𝑥} ≠ ∅
3	2	nesymi 2995	. . . 4 ⊢ ¬ ∅ = {𝑥}
4	3	nex 1795	. . 3 ⊢ ¬ ∃𝑥∅ = {𝑥}
5		bj-elsngl 36447	. . . 4 ⊢ (∅ ∈ sngl 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∅ = {𝑥})
6		rexex 3073	. . . 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 3046	1 ⊢ ∅ ∉ sngl 𝐴

Syntax hints:   = wceq 1534  ∃wex 1774   ∈ wcel 2099   ∉ wnel 3043  ∃wrex 3067  ∅c0 4323  {csn 4629  sngl bj-csngl 36444

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-nel 3044  df-rex 3068  df-v 3473  df-dif 3950  df-un 3952  df-nul 4324  df-sn 4630  df-pr 4632  df-bj-sngl 36445

This theorem is referenced by: (None)