Theorem bj-axbun 37101

Description: Two ways of stating the axiom of binary union (which is the universal closure of either side, see ax-bj-bun 37102). (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 4102	. 2 ⊢ (𝑡 ∈ (𝑥 ∪ 𝑦) ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 ∈ 𝑦))
2	1	bj-clex 37096	1 ⊢ ((𝑥 ∪ 𝑦) ∈ V ↔ ∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 ∈ 𝑦)))

Syntax hints:   ↔ wb 206   ∨ wo 847  ∀wal 1539  ∃wex 1780   ∈ wcel 2113  Vcvv 3437   ∪ cun 3896

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-v 3439  df-un 3903

This theorem is referenced by:  bj-unexg  37103