Metamath Proof Explorer
Description: Restricted class abstraction with true formula. (Contributed by BJ, 22-Apr-2019)
|
|
Ref |
Expression |
|
Assertion |
bj-rabtr |
⊢ { 𝑥 ∈ 𝐴 ∣ ⊤ } = 𝐴 |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
ssrab2 |
⊢ { 𝑥 ∈ 𝐴 ∣ ⊤ } ⊆ 𝐴 |
2 |
|
ssid |
⊢ 𝐴 ⊆ 𝐴 |
3 |
|
tru |
⊢ ⊤ |
4 |
3
|
rgenw |
⊢ ∀ 𝑥 ∈ 𝐴 ⊤ |
5 |
|
ssrab |
⊢ ( 𝐴 ⊆ { 𝑥 ∈ 𝐴 ∣ ⊤ } ↔ ( 𝐴 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 ⊤ ) ) |
6 |
2 4 5
|
mpbir2an |
⊢ 𝐴 ⊆ { 𝑥 ∈ 𝐴 ∣ ⊤ } |
7 |
1 6
|
eqssi |
⊢ { 𝑥 ∈ 𝐴 ∣ ⊤ } = 𝐴 |