Theorem bj-0nelsngl 33481

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 7826). (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 3417	. . . . . 6 ⊢ 𝑥 ∈ V
2	1	snnz 4528	. . . . 5 ⊢ {𝑥} ≠ ∅
3	2	nesymi 3056	. . . 4 ⊢ ¬ ∅ = {𝑥}
4	3	nex 1901	. . 3 ⊢ ¬ ∃𝑥∅ = {𝑥}
5		bj-elsngl 33478	. . . 4 ⊢ (∅ ∈ sngl 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∅ = {𝑥})
6		rexex 3210	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∅ = {𝑥} → ∃𝑥∅ = {𝑥})
7	5, 6	sylbi 209	. . 3 ⊢ (∅ ∈ sngl 𝐴 → ∃𝑥∅ = {𝑥})
8	4, 7	mto 189	. 2 ⊢ ¬ ∅ ∈ sngl 𝐴
9	8	nelir 3105	1 ⊢ ∅ ∉ sngl 𝐴

Syntax hints:   = wceq 1658  ∃wex 1880   ∈ wcel 2166   ∉ wnel 3102  ∃wrex 3118  ∅c0 4144  {csn 4397  sngl bj-csngl 33475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-v 3416  df-dif 3801  df-un 3803  df-nul 4145  df-sn 4398  df-pr 4400  df-bj-sngl 33476

This theorem is referenced by: (None)