Theorem hash1elsn

Description: A set of size 1 with a known element is the singleton of that element. (Contributed by Rohan Ridenour, 3-Aug-2023)

	Ref	Expression
Hypotheses	hash1elsn.1	|- ( ph -> ( # ` A ) = 1 )
	hash1elsn.2	|- ( ph -> B e. A )
	hash1elsn.3	|- ( ph -> A e. V )
Assertion	hash1elsn	|- ( ph -> A = { B } )

Proof

Step	Hyp	Ref	Expression
1		hash1elsn.1	|- ( ph -> ( # ` A ) = 1 )
2		hash1elsn.2	|- ( ph -> B e. A )
3		hash1elsn.3	|- ( ph -> A e. V )
4		hashen1	|- ( A e. V -> ( ( # ` A ) = 1 <-> A ~~ 1o ) )
5	3 4	syl	|- ( ph -> ( ( # ` A ) = 1 <-> A ~~ 1o ) )
6	1 5	mpbid	|- ( ph -> A ~~ 1o )
7		en1	|- ( A ~~ 1o <-> E. x A = { x } )
8	6 7	sylib	|- ( ph -> E. x A = { x } )
9		simpr	|- ( ( ph /\ A = { x } ) -> A = { x } )
10	2	adantr	|- ( ( ph /\ A = { x } ) -> B e. A )
11	10 9	eleqtrd	|- ( ( ph /\ A = { x } ) -> B e. { x } )
12		elsni	|- ( B e. { x } -> B = x )
13	11 12	syl	|- ( ( ph /\ A = { x } ) -> B = x )
14	13	sneqd	|- ( ( ph /\ A = { x } ) -> { B } = { x } )
15	9 14	eqtr4d	|- ( ( ph /\ A = { x } ) -> A = { B } )
16	8 15	exlimddv	|- ( ph -> A = { B } )