Theorem snssb

Description: Characterization of the inclusion of a singleton in a class. (Contributed by BJ, 1-Jan-2025)

	Ref	Expression
Assertion	snssb	$$⊢\left\{A\right\}\subseteq B\leftrightarrow \left(A\in \mathrm{V}\to A\in B\right)$$

Proof

Step	Hyp	Ref	Expression
1		df-ss	$$⊢\left\{A\right\}\subseteq B\leftrightarrow \forall x\left(x\in \left\{A\right\}\to x\in B\right)$$
2		velsn	$$⊢x\in \left\{A\right\}\leftrightarrow x=A$$
3	2	imbi1i	$$⊢\left(x\in \left\{A\right\}\to x\in B\right)\leftrightarrow \left(x=A\to x\in B\right)$$
4	3	albii	$$⊢\forall x\left(x\in \left\{A\right\}\to x\in B\right)\leftrightarrow \forall x\left(x=A\to x\in B\right)$$
5		eleq1	$$⊢x=A\to \left(x\in B\leftrightarrow A\in B\right)$$
6	5	pm5.74i	$$⊢\left(x=A\to x\in B\right)\leftrightarrow \left(x=A\to A\in B\right)$$
7	6	albii	$$⊢\forall x\left(x=A\to x\in B\right)\leftrightarrow \forall x\left(x=A\to A\in B\right)$$
8		19.23v	$$⊢\forall x\left(x=A\to A\in B\right)\leftrightarrow \left(\exists xx=A\to A\in B\right)$$
9		isset	$$⊢A\in \mathrm{V}\leftrightarrow \exists xx=A$$
10	9	bicomi	$$⊢\exists xx=A\leftrightarrow A\in \mathrm{V}$$
11	10	imbi1i	$$⊢\left(\exists xx=A\to A\in B\right)\leftrightarrow \left(A\in \mathrm{V}\to A\in B\right)$$
12	7 8 11	3bitri	$$⊢\forall x\left(x=A\to x\in B\right)\leftrightarrow \left(A\in \mathrm{V}\to A\in B\right)$$
13	1 4 12	3bitri	$$⊢\left\{A\right\}\subseteq B\leftrightarrow \left(A\in \mathrm{V}\to A\in B\right)$$