Theorem rabtru

Description: Abstract builder using the constant wff T. . (Contributed by Thierry Arnoux, 4-May-2020)

	Ref	Expression
Hypothesis	rabtru.1	⊢ Ⅎ 𝑥 𝐴
Assertion	rabtru	⊢ { 𝑥 ∈ 𝐴 ∣ ⊤ } = 𝐴

Proof

Step	Hyp	Ref	Expression
1		rabtru.1	⊢ Ⅎ 𝑥 𝐴
2		tru	⊢ ⊤
3		nfcv	⊢ Ⅎ 𝑥 𝑦
4		nftru	⊢ Ⅎ 𝑥 ⊤
5		biidd	⊢ ( 𝑥 = 𝑦 → ( ⊤ ↔ ⊤ ) )
6	3 1 4 5	elrabf	⊢ ( 𝑦 ∈ { 𝑥 ∈ 𝐴 ∣ ⊤ } ↔ ( 𝑦 ∈ 𝐴 ∧ ⊤ ) )
7	2 6	mpbiran2	⊢ ( 𝑦 ∈ { 𝑥 ∈ 𝐴 ∣ ⊤ } ↔ 𝑦 ∈ 𝐴 )
8	7	eqriv	⊢ { 𝑥 ∈ 𝐴 ∣ ⊤ } = 𝐴