Theorem bj-0nelsngl 33084

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 7605). (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 3234	. . . . . 6 ⊢ 𝑥 ∈ V
2	1	snnz 4340	. . . . 5 ⊢ {𝑥} ≠ ∅
3	2	nesymi 2880	. . . 4 ⊢ ¬ ∅ = {𝑥}
4	3	nex 1771	. . 3 ⊢ ¬ ∃𝑥∅ = {𝑥}
5		bj-elsngl 33081	. . . 4 ⊢ (∅ ∈ sngl 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∅ = {𝑥})
6		rexex 3031	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∅ = {𝑥} → ∃𝑥∅ = {𝑥})
7	5, 6	sylbi 207	. . 3 ⊢ (∅ ∈ sngl 𝐴 → ∃𝑥∅ = {𝑥})
8	4, 7	mto 188	. 2 ⊢ ¬ ∅ ∈ sngl 𝐴
9	8	nelir 2929	1 ⊢ ∅ ∉ sngl 𝐴

Syntax hints:   = wceq 1523  ∃wex 1744   ∈ wcel 2030   ∉ wnel 2926  ∃wrex 2942  ∅c0 3948  {csn 4210  sngl bj-csngl 33078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-v 3233  df-dif 3610  df-un 3612  df-nul 3949  df-sn 4211  df-pr 4213  df-bj-sngl 33079

This theorem is referenced by: (None)