Theorem bj-0nelsngl 37635

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 8451). (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 3458	. . . . . 6 ⊢ 𝑥 ∈ V
2	1	snnz 4741	. . . . 5 ⊢ {𝑥} ≠ ∅
3	2	nesymi 3014	. . . 4 ⊢ ¬ ∅ = {𝑥}
4	3	nex 1829	. . 3 ⊢ ¬ ∃𝑥∅ = {𝑥}
5		bj-elsngl 37632	. . . 4 ⊢ (∅ ∈ sngl 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∅ = {𝑥})
6		rexex 3094	. . . 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 3066	1 ⊢ ∅ ∉ sngl 𝐴

This proof depends on syntax axioms:   = wceq 1569  ∃wex 1808   ∈ wcel 2142   ∉ wnel 3063  ∃wrex 3088  ∅c0 4285  {csn 4588  sngl bj-csngl 37629

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-nel 3064  df-rex 3089  df-v 3456  df-dif 3907  df-un 3909  df-nul 4286  df-sn 4589  df-pr 4591  df-bj-sngl 37630

This theorem is used by: (None)