Theorem bj-adjfrombun 37080

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 3440	. 2 ⊢ 𝑥 ∈ V
2		bj-snexg 37068	. . 3 ⊢ (𝑦 ∈ V → {𝑦} ∈ V)
3	2	elv 3441	. 2 ⊢ {𝑦} ∈ V
4		bj-unexg 37072	. 2 ⊢ ((𝑥 ∈ V ∧ {𝑦} ∈ V) → (𝑥 ∪ {𝑦}) ∈ V)
5	1, 3, 4	mp2an 692	1 ⊢ (𝑥 ∪ {𝑦}) ∈ V

Syntax hints:   ∈ wcel 2111  Vcvv 3436   ∪ cun 3895  {csn 4571

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 2113  ax-9 2121  ax-12 2180  ax-ext 2703  ax-bj-sn 37067  ax-bj-bun 37071

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 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-un 3902  df-sn 4572

This theorem is referenced by: (None)