Theorem bj-0nelsngl 36937

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 8522). (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 3492	. . . . . 6 ⊢ 𝑥 ∈ V
2	1	snnz 4801	. . . . 5 ⊢ {𝑥} ≠ ∅
3	2	nesymi 3004	. . . 4 ⊢ ¬ ∅ = {𝑥}
4	3	nex 1798	. . 3 ⊢ ¬ ∃𝑥∅ = {𝑥}
5		bj-elsngl 36934	. . . 4 ⊢ (∅ ∈ sngl 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∅ = {𝑥})
6		rexex 3082	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∅ = {𝑥} → ∃𝑥∅ = {𝑥})
7	5, 6	sylbi 217	. . 3 ⊢ (∅ ∈ sngl 𝐴 → ∃𝑥∅ = {𝑥})
8	4, 7	mto 197	. 2 ⊢ ¬ ∅ ∈ sngl 𝐴
9	8	nelir 3055	1 ⊢ ∅ ∉ sngl 𝐴

Syntax hints:   = wceq 1537  ∃wex 1777   ∈ wcel 2108   ∉ wnel 3052  ∃wrex 3076  ∅c0 4352  {csn 4648  sngl bj-csngl 36931

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-nel 3053  df-rex 3077  df-v 3490  df-dif 3979  df-un 3981  df-nul 4353  df-sn 4649  df-pr 4651  df-bj-sngl 36932

This theorem is referenced by: (None)