Theorem bj-denoteslem

Description: Lemma for bj-denotes . (Contributed by BJ, 24-Apr-2024) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-denoteslem	⊢ ( ∃ 𝑥 𝑥 = 𝐴 ↔ 𝐴 ∈ { 𝑦 ∣ ⊤ } )

Proof

Step	Hyp	Ref	Expression
1		vextru	⊢ 𝑥 ∈ { 𝑦 ∣ ⊤ }
2	1	biantru	⊢ ( 𝑥 = 𝐴 ↔ ( 𝑥 = 𝐴 ∧ 𝑥 ∈ { 𝑦 ∣ ⊤ } ) )
3	2	exbii	⊢ ( ∃ 𝑥 𝑥 = 𝐴 ↔ ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝑥 ∈ { 𝑦 ∣ ⊤ } ) )
4		dfclel	⊢ ( 𝐴 ∈ { 𝑦 ∣ ⊤ } ↔ ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝑥 ∈ { 𝑦 ∣ ⊤ } ) )
5	3 4	bitr4i	⊢ ( ∃ 𝑥 𝑥 = 𝐴 ↔ 𝐴 ∈ { 𝑦 ∣ ⊤ } )