Theorem bj-0nelsngl

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 ). (Contributed by BJ, 6-Oct-2018)

	Ref	Expression
Assertion	bj-0nelsngl	⊢ ∅ ∉ sngl 𝐴

Proof

Step	Hyp	Ref	Expression
1		vex	⊢ 𝑥 ∈ V
2	1	snnz	⊢ { 𝑥 } ≠ ∅
3	2	nesymi	⊢ ¬ ∅ = { 𝑥 }
4	3	nex	⊢ ¬ ∃ 𝑥 ∅ = { 𝑥 }
5		bj-elsngl	⊢ ( ∅ ∈ sngl 𝐴 ↔ ∃ 𝑥 ∈ 𝐴 ∅ = { 𝑥 } )
6		rexex	⊢ ( ∃ 𝑥 ∈ 𝐴 ∅ = { 𝑥 } → ∃ 𝑥 ∅ = { 𝑥 } )
7	5 6	sylbi	⊢ ( ∅ ∈ sngl 𝐴 → ∃ 𝑥 ∅ = { 𝑥 } )
8	4 7	mto	⊢ ¬ ∅ ∈ sngl 𝐴
9	8	nelir	⊢ ∅ ∉ sngl 𝐴