Theorem bj-0nelsngl 34407

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 8085). (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 3444	. . . . . 6 ⊢ 𝑥 ∈ V
2	1	snnz 4672	. . . . 5 ⊢ {𝑥} ≠ ∅
3	2	nesymi 3044	. . . 4 ⊢ ¬ ∅ = {𝑥}
4	3	nex 1802	. . 3 ⊢ ¬ ∃𝑥∅ = {𝑥}
5		bj-elsngl 34404	. . . 4 ⊢ (∅ ∈ sngl 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∅ = {𝑥})
6		rexex 3203	. . . 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 3094	1 ⊢ ∅ ∉ sngl 𝐴

Syntax hints:   = wceq 1538  ∃wex 1781   ∈ wcel 2111   ∉ wnel 3091  ∃wrex 3107  ∅c0 4243  {csn 4525  sngl bj-csngl 34401

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ne 2988  df-nel 3092  df-rex 3112  df-v 3443  df-dif 3884  df-un 3886  df-nul 4244  df-sn 4526  df-pr 4528  df-bj-sngl 34402

This theorem is referenced by: (None)