Theorem bj-rabtrAUTO

Description: Proof of bj-rabtr found automatically by the Metamath program "MM-PA> IMPROVE ALL / DEPTH 3 / 3" command followed by "MM-PA> MINIMIZE__WITH *". (Contributed by BJ, 22-Apr-2019) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	bj-rabtrAUTO	⊢ { 𝑥 ∈ 𝐴 ∣ ⊤ } = 𝐴

Proof

Step	Hyp	Ref	Expression
1		ssrab2	⊢ { 𝑥 ∈ 𝐴 ∣ ⊤ } ⊆ 𝐴
2		ssid	⊢ 𝐴 ⊆ 𝐴
3	2	a1i	⊢ ( ⊤ → 𝐴 ⊆ 𝐴 )
4		simpl	⊢ ( ( ⊤ ∧ 𝑥 ∈ 𝐴 ) → ⊤ )
5	3 4	ssrabdv	⊢ ( ⊤ → 𝐴 ⊆ { 𝑥 ∈ 𝐴 ∣ ⊤ } )
6	5	mptru	⊢ 𝐴 ⊆ { 𝑥 ∈ 𝐴 ∣ ⊤ }
7	1 6	eqssi	⊢ { 𝑥 ∈ 𝐴 ∣ ⊤ } = 𝐴