Theorem bj-0nelsngl 34287

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 8105). (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 3500	. . . . . 6 ⊢ 𝑥 ∈ V
2	1	snnz 4714	. . . . 5 ⊢ {𝑥} ≠ ∅
3	2	nesymi 3076	. . . 4 ⊢ ¬ ∅ = {𝑥}
4	3	nex 1800	. . 3 ⊢ ¬ ∃𝑥∅ = {𝑥}
5		bj-elsngl 34284	. . . 4 ⊢ (∅ ∈ sngl 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∅ = {𝑥})
6		rexex 3243	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∅ = {𝑥} → ∃𝑥∅ = {𝑥})
7	5, 6	sylbi 219	. . 3 ⊢ (∅ ∈ sngl 𝐴 → ∃𝑥∅ = {𝑥})
8	4, 7	mto 199	. 2 ⊢ ¬ ∅ ∈ sngl 𝐴
9	8	nelir 3129	1 ⊢ ∅ ∉ sngl 𝐴

Syntax hints:   = wceq 1536  ∃wex 1779   ∈ wcel 2113   ∉ wnel 3126  ∃wrex 3142  ∅c0 4294  {csn 4570  sngl bj-csngl 34281

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-nel 3127  df-rex 3147  df-v 3499  df-dif 3942  df-un 3944  df-nul 4295  df-sn 4571  df-pr 4573  df-bj-sngl 34282

This theorem is referenced by: (None)