Theorem bj-0nelsngl 37884

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 8476). (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 3454	. . . . . 6 ⊢ 𝑥 ∈ V
2	1	snnz 4736	. . . . 5 ⊢ {𝑥} ≠ ∅
3	2	nesymi 3012	. . . 4 ⊢ ¬ ∅ = {𝑥}
4	3	nex 1832	. . 3 ⊢ ¬ ∃𝑥∅ = {𝑥}
5		bj-elsngl 37881	. . . 4 ⊢ (∅ ∈ sngl 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∅ = {𝑥})
6		rexex 3092	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∅ = {𝑥} → ∃𝑥∅ = {𝑥})
7	5, 6	sylbi 220	. . 3 ⊢ (∅ ∈ sngl 𝐴 → ∃𝑥∅ = {𝑥})
8	4, 7	mto 200	. 2 ⊢ ¬ ∅ ∈ sngl 𝐴
9	8	nelir 3064	1 ⊢ ∅ ∉ sngl 𝐴

This proof depends on syntax axioms:   = wceq 1569  ∃wex 1811   ∈ wcel 2144   ∉ wnel 3061  ∃wrex 3086  ∅c0 4278  {csn 4583  sngl bj-csngl 37878

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-nel 3062  df-rex 3087  df-v 3452  df-dif 4047  df-un 4151  df-nul 4279  df-sn 4584  df-pr 4586  df-bj-sngl 37879

This theorem is used by: (None)