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 8449). (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 3459	. . . . . 6 ⊢ 𝑥 ∈ V
2	1	snnz 4742	. . . . 5 ⊢ {𝑥} ≠ ∅
3	2	nesymi 3015	. . . 4 ⊢ ¬ ∅ = {𝑥}
4	3	nex 1830	. . 3 ⊢ ¬ ∃𝑥∅ = {𝑥}
5		bj-elsngl 37632	. . . 4 ⊢ (∅ ∈ sngl 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∅ = {𝑥})
6		rexex 3095	. . . 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 3067	1 ⊢ ∅ ∉ sngl 𝐴

This proof depends on syntax axioms:   = wceq 1570  ∃wex 1809   ∈ wcel 2143   ∉ wnel 3064  ∃wrex 3089  ∅c0 4286  {csn 4589  sngl bj-csngl 37629

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-pr 5404

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-nel 3065  df-rex 3090  df-v 3457  df-dif 3908  df-un 3910  df-nul 4287  df-sn 4590  df-pr 4592  df-bj-sngl 37630

This theorem is used by: (None)