Theorem snssg

Description: The singleton of an element of a class is a subset of the class (general form of snss ). Theorem 7.4 of Quine p. 49. (Contributed by NM, 22-Jul-2001)

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

Proof

Step	Hyp	Ref	Expression
1		velsn	$$⊢x\in \left\{A\right\}\leftrightarrow x=A$$
2	1	imbi1i	$$⊢\left(x\in \left\{A\right\}\to x\in B\right)\leftrightarrow \left(x=A\to x\in B\right)$$
3	2	albii	$$⊢\forall x\left(x\in \left\{A\right\}\to x\in B\right)\leftrightarrow \forall x\left(x=A\to x\in B\right)$$
4	3	a1i	$$⊢A\in V\to \left(\forall x\left(x\in \left\{A\right\}\to x\in B\right)\leftrightarrow \forall x\left(x=A\to x\in B\right)\right)$$
5		dfss2	$$⊢\left\{A\right\}\subseteq B\leftrightarrow \forall x\left(x\in \left\{A\right\}\to x\in B\right)$$
6	5	a1i	$$⊢A\in V\to \left(\left\{A\right\}\subseteq B\leftrightarrow \forall x\left(x\in \left\{A\right\}\to x\in B\right)\right)$$
7		clel2g	$$⊢A\in V\to \left(A\in B\leftrightarrow \forall x\left(x=A\to x\in B\right)\right)$$
8	4 6 7	3bitr4rd	$$⊢A\in V\to \left(A\in B\leftrightarrow \left\{A\right\}\subseteq B\right)$$