Theorem rabsnel

Description: Truth implied by equality of a restricted class abstraction and a singleton. (Contributed by Thierry Arnoux, 15-Sep-2018)

	Ref	Expression
Hypothesis	rabsnel.1	⊢ 𝐵 ∈ V
Assertion	rabsnel	⊢ ( { 𝑥 ∈ 𝐴 ∣ 𝜑 } = { 𝐵 } → 𝐵 ∈ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		rabsnel.1	⊢ 𝐵 ∈ V
2	1	snid	⊢ 𝐵 ∈ { 𝐵 }
3		eleq2	⊢ ( { 𝑥 ∈ 𝐴 ∣ 𝜑 } = { 𝐵 } → ( 𝐵 ∈ { 𝑥 ∈ 𝐴 ∣ 𝜑 } ↔ 𝐵 ∈ { 𝐵 } ) )
4	2 3	mpbiri	⊢ ( { 𝑥 ∈ 𝐴 ∣ 𝜑 } = { 𝐵 } → 𝐵 ∈ { 𝑥 ∈ 𝐴 ∣ 𝜑 } )
5		elrabi	⊢ ( 𝐵 ∈ { 𝑥 ∈ 𝐴 ∣ 𝜑 } → 𝐵 ∈ 𝐴 )
6	4 5	syl	⊢ ( { 𝑥 ∈ 𝐴 ∣ 𝜑 } = { 𝐵 } → 𝐵 ∈ 𝐴 )