Theorem cardsn

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

	Ref	Expression
Assertion	cardsn	⊢ ( 𝐴 ∈ 𝑉 → ( card ‘ { 𝐴 } ) = 1o )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ { 𝐴 } = { 𝐴 }
2		sneq	⊢ ( 𝑥 = 𝐴 → { 𝑥 } = { 𝐴 } )
3	2	eqeq2d	⊢ ( 𝑥 = 𝐴 → ( { 𝐴 } = { 𝑥 } ↔ { 𝐴 } = { 𝐴 } ) )
4	3	spcegv	⊢ ( 𝐴 ∈ 𝑉 → ( { 𝐴 } = { 𝐴 } → ∃ 𝑥 { 𝐴 } = { 𝑥 } ) )
5	1 4	mpi	⊢ ( 𝐴 ∈ 𝑉 → ∃ 𝑥 { 𝐴 } = { 𝑥 } )
6		card1	⊢ ( ( card ‘ { 𝐴 } ) = 1o ↔ ∃ 𝑥 { 𝐴 } = { 𝑥 } )
7	5 6	sylibr	⊢ ( 𝐴 ∈ 𝑉 → ( card ‘ { 𝐴 } ) = 1o )