Theorem bj-elabtru

Description: This is as close as we can get to proving extensionality for "the" "universal" class without ax-ext . (Contributed by BJ, 24-Apr-2024)

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

Proof

Step	Hyp	Ref	Expression
1		bj-denoteslem	⊢ ( ∃ 𝑧 𝑧 = 𝐴 ↔ 𝐴 ∈ { 𝑥 ∣ ⊤ } )
2		bj-denoteslem	⊢ ( ∃ 𝑧 𝑧 = 𝐴 ↔ 𝐴 ∈ { 𝑦 ∣ ⊤ } )
3	1 2	bitr3i	⊢ ( 𝐴 ∈ { 𝑥 ∣ ⊤ } ↔ 𝐴 ∈ { 𝑦 ∣ ⊤ } )