Theorem bj-elabtru 37475

Description: This is as close as we can get to proving extensionality for "the" "universal" class without ax-ext 2733. (Contributed by BJ, 24-Apr-2024.) (Proof modification is discouraged.)

Assertion

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

Proof of Theorem bj-elabtru

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		issettru 2839	. 2 ⊢ (∃𝑧 𝑧 = 𝐴 ↔ 𝐴 ∈ {𝑥 ∣ ⊤})
2		issettru 2839	. 2 ⊢ (∃𝑧 𝑧 = 𝐴 ↔ 𝐴 ∈ {𝑦 ∣ ⊤})
3	1, 2	bitr3i 280	1 ⊢ (𝐴 ∈ {𝑥 ∣ ⊤} ↔ 𝐴 ∈ {𝑦 ∣ ⊤})

Syntax hints:   ↔ wb 209   = wceq 1568  ⊤wtru 1569  ∃wex 1807   ∈ wcel 2141  {cab 2739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1571  df-ex 1808  df-sb 2095  df-clab 2740  df-clel 2836

This theorem is referenced by: (None)