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	⊢ ( 𝜑 → ( ♯ ‘ 𝐴 ) = 1 )
	hash1elsn.2	⊢ ( 𝜑 → 𝐵 ∈ 𝐴 )
	hash1elsn.3	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
Assertion	hash1elsn	⊢ ( 𝜑 → 𝐴 = { 𝐵 } )

Proof

Step	Hyp	Ref	Expression
1		hash1elsn.1	⊢ ( 𝜑 → ( ♯ ‘ 𝐴 ) = 1 )
2		hash1elsn.2	⊢ ( 𝜑 → 𝐵 ∈ 𝐴 )
3		hash1elsn.3	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
4		hashen1	⊢ ( 𝐴 ∈ 𝑉 → ( ( ♯ ‘ 𝐴 ) = 1 ↔ 𝐴 ≈ 1o ) )
5	3 4	syl	⊢ ( 𝜑 → ( ( ♯ ‘ 𝐴 ) = 1 ↔ 𝐴 ≈ 1o ) )
6	1 5	mpbid	⊢ ( 𝜑 → 𝐴 ≈ 1o )
7		en1	⊢ ( 𝐴 ≈ 1o ↔ ∃ 𝑥 𝐴 = { 𝑥 } )
8	6 7	sylib	⊢ ( 𝜑 → ∃ 𝑥 𝐴 = { 𝑥 } )
9		simpr	⊢ ( ( 𝜑 ∧ 𝐴 = { 𝑥 } ) → 𝐴 = { 𝑥 } )
10	2	adantr	⊢ ( ( 𝜑 ∧ 𝐴 = { 𝑥 } ) → 𝐵 ∈ 𝐴 )
11	10 9	eleqtrd	⊢ ( ( 𝜑 ∧ 𝐴 = { 𝑥 } ) → 𝐵 ∈ { 𝑥 } )
12		elsni	⊢ ( 𝐵 ∈ { 𝑥 } → 𝐵 = 𝑥 )
13	11 12	syl	⊢ ( ( 𝜑 ∧ 𝐴 = { 𝑥 } ) → 𝐵 = 𝑥 )
14	13	sneqd	⊢ ( ( 𝜑 ∧ 𝐴 = { 𝑥 } ) → { 𝐵 } = { 𝑥 } )
15	9 14	eqtr4d	⊢ ( ( 𝜑 ∧ 𝐴 = { 𝑥 } ) → 𝐴 = { 𝐵 } )
16	8 15	exlimddv	⊢ ( 𝜑 → 𝐴 = { 𝐵 } )