Theorem bj-adjfrombun 36231

Description: Adjunction from singleton and binary union. (Contributed by BJ, 19-Jan-2025.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-adjfrombun	⊢ (𝑥 ∪ {𝑦}) ∈ V

Proof of Theorem bj-adjfrombun

Step	Hyp	Ref	Expression
1		vex 3477	. 2 ⊢ 𝑥 ∈ V
2		bj-snexg 36219	. . 3 ⊢ (𝑦 ∈ V → {𝑦} ∈ V)
3	2	elv 3479	. 2 ⊢ {𝑦} ∈ V
4		bj-unexg 36223	. 2 ⊢ ((𝑥 ∈ V ∧ {𝑦} ∈ V) → (𝑥 ∪ {𝑦}) ∈ V)
5	1, 3, 4	mp2an 689	1 ⊢ (𝑥 ∪ {𝑦}) ∈ V

Syntax hints:   ∈ wcel 2105  Vcvv 3473   ∪ cun 3946  {csn 4628

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-12 2170  ax-ext 2702  ax-bj-sn 36218  ax-bj-bun 36222

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-un 3953  df-sn 4629

This theorem is referenced by: (None)