Theorem bj-axbun 37650

Description: Two ways of stating the axiom of binary union (which is the universal closure of either side, see ax-bj-bun 37651). (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 4108	. 2 ⊢ (𝑡 ∈ (𝑥 ∪ 𝑦) ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 ∈ 𝑦))
2	1	bj-clex 37645	1 ⊢ ((𝑥 ∪ 𝑦) ∈ V ↔ ∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 ∈ 𝑦)))

Syntax hints:   ↔ wb 209   ∨ wo 860  ∀wal 1568  ∃wex 1809   ∈ wcel 2143  Vcvv 3455   ∪ cun 3904

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-un 3911

This theorem is referenced by:  bj-unexg  37652