Theorem cardsn

Description: A singleton has cardinality one. (Contributed by Mario Carneiro, 10-Jan-2013)

	Ref	Expression
Assertion	cardsn	$$⊢A\in V\to \mathrm{card}\left(\left\{A\right\}\right)=1_{𝑜}$$

Proof

Step	Hyp	Ref	Expression
1		eqid	$$⊢\left\{A\right\}=\left\{A\right\}$$
2		sneq	$$⊢x=A\to \left\{x\right\}=\left\{A\right\}$$
3	2	eqeq2d	$$⊢x=A\to \left(\left\{A\right\}=\left\{x\right\}\leftrightarrow \left\{A\right\}=\left\{A\right\}\right)$$
4	3	spcegv	$$⊢A\in V\to \left(\left\{A\right\}=\left\{A\right\}\to \exists x\left\{A\right\}=\left\{x\right\}\right)$$
5	1 4	mpi	$$⊢A\in V\to \exists x\left\{A\right\}=\left\{x\right\}$$
6		card1	$$⊢\mathrm{card}\left(\left\{A\right\}\right)=1_{𝑜}\leftrightarrow \exists x\left\{A\right\}=\left\{x\right\}$$
7	5 6	sylibr	$$⊢A\in V\to \mathrm{card}\left(\left\{A\right\}\right)=1_{𝑜}$$