Theorem bj-elabtru 34204

Description: This is as close as we can get to proving extensionality for "the" "universal" class without ax-ext 2793. (Contributed by BJ, 24-Apr-2024.)

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		bj-denoteslem 34201	. 2 ⊢ (∃𝑧 𝑧 = 𝐴 ↔ 𝐴 ∈ {𝑥 ∣ ⊤})
2		bj-denoteslem 34201	. 2 ⊢ (∃𝑧 𝑧 = 𝐴 ↔ 𝐴 ∈ {𝑦 ∣ ⊤})
3	1, 2	bitr3i 279	1 ⊢ (𝐴 ∈ {𝑥 ∣ ⊤} ↔ 𝐴 ∈ {𝑦 ∣ ⊤})

Syntax hints:   ↔ wb 208   = wceq 1537  ⊤wtru 1538  ∃wex 1780   ∈ wcel 2114  {cab 2799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116

This theorem depends on definitions:  df-bi 209  df-an 399  df-tru 1540  df-ex 1781  df-sb 2070  df-clab 2800  df-clel 2893

This theorem is referenced by: (None)