Theorem bj-0nelsngl 35161

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 8297). (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 3436	. . . . . 6 ⊢ 𝑥 ∈ V
2	1	snnz 4712	. . . . 5 ⊢ {𝑥} ≠ ∅
3	2	nesymi 3001	. . . 4 ⊢ ¬ ∅ = {𝑥}
4	3	nex 1803	. . 3 ⊢ ¬ ∃𝑥∅ = {𝑥}
5		bj-elsngl 35158	. . . 4 ⊢ (∅ ∈ sngl 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∅ = {𝑥})
6		rexex 3171	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∅ = {𝑥} → ∃𝑥∅ = {𝑥})
7	5, 6	sylbi 216	. . 3 ⊢ (∅ ∈ sngl 𝐴 → ∃𝑥∅ = {𝑥})
8	4, 7	mto 196	. 2 ⊢ ¬ ∅ ∈ sngl 𝐴
9	8	nelir 3052	1 ⊢ ∅ ∉ sngl 𝐴

Syntax hints:   = wceq 1539  ∃wex 1782   ∈ wcel 2106   ∉ wnel 3049  ∃wrex 3065  ∅c0 4256  {csn 4561  sngl bj-csngl 35155

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-nel 3050  df-rex 3070  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-sn 4562  df-pr 4564  df-bj-sngl 35156

This theorem is referenced by: (None)