Theorem bj-elabtru 36834

Description: This is as close as we can get to proving extensionality for "the" "universal" class without ax-ext 2706. (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 2811	. 2 ⊢ (∃𝑧 𝑧 = 𝐴 ↔ 𝐴 ∈ {𝑥 ∣ ⊤})
2		issettru 2811	. 2 ⊢ (∃𝑧 𝑧 = 𝐴 ↔ 𝐴 ∈ {𝑦 ∣ ⊤})
3	1, 2	bitr3i 277	1 ⊢ (𝐴 ∈ {𝑥 ∣ ⊤} ↔ 𝐴 ∈ {𝑦 ∣ ⊤})

Syntax hints:   ↔ wb 206   = wceq 1539  ⊤wtru 1540  ∃wex 1778   ∈ wcel 2107  {cab 2712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2713  df-clel 2808

This theorem is referenced by: (None)