Metamath Proof Explorer


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) (Proof modification is discouraged.)

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

Proof

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