Theorem bj-axbun 35707

Description: Two ways of stating the axiom of binary union (which is the universal closure of either side, see ax-bj-bun 35708). (Contributed by BJ, 12-Jan-2025.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-axbun	⊢ ((𝑥 ∪ 𝑦) ∈ V ↔ ∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 ∈ 𝑦)))

Distinct variable groups:   𝑥,𝑧,𝑡   𝑦,𝑧,𝑡

Proof of Theorem bj-axbun

Step	Hyp	Ref	Expression
1		elun 4143	. 2 ⊢ (𝑡 ∈ (𝑥 ∪ 𝑦) ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 ∈ 𝑦))
2	1	bj-clex 35702	1 ⊢ ((𝑥 ∪ 𝑦) ∈ V ↔ ∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 ∈ 𝑦)))

Syntax hints:   ↔ wb 205   ∨ wo 845  ∀wal 1539  ∃wex 1781   ∈ wcel 2106  Vcvv 3472   ∪ cun 3941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3474  df-un 3948

This theorem is referenced by:  bj-unexg  35709