Metamath Proof Explorer


Theorem bj-rabtr

Description: Restricted class abstraction with true formula. (Contributed by BJ, 22-Apr-2019)

Ref Expression
Assertion bj-rabtr { 𝑥𝐴 ∣ ⊤ } = 𝐴

Proof

Step Hyp Ref Expression
1 ssrab2 { 𝑥𝐴 ∣ ⊤ } ⊆ 𝐴
2 ssid 𝐴𝐴
3 tru
4 3 rgenw 𝑥𝐴
5 ssrab ( 𝐴 ⊆ { 𝑥𝐴 ∣ ⊤ } ↔ ( 𝐴𝐴 ∧ ∀ 𝑥𝐴 ⊤ ) )
6 2 4 5 mpbir2an 𝐴 ⊆ { 𝑥𝐴 ∣ ⊤ }
7 1 6 eqssi { 𝑥𝐴 ∣ ⊤ } = 𝐴