Theorem bj-adjfrombun 37028

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 3481	. 2 ⊢ 𝑥 ∈ V
2		bj-snexg 37016	. . 3 ⊢ (𝑦 ∈ V → {𝑦} ∈ V)
3	2	elv 3482	. 2 ⊢ {𝑦} ∈ V
4		bj-unexg 37020	. 2 ⊢ ((𝑥 ∈ V ∧ {𝑦} ∈ V) → (𝑥 ∪ {𝑦}) ∈ V)
5	1, 3, 4	mp2an 692	1 ⊢ (𝑥 ∪ {𝑦}) ∈ V

Syntax hints:   ∈ wcel 2105  Vcvv 3477   ∪ cun 3960  {csn 4630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-12 2174  ax-ext 2705  ax-bj-sn 37015  ax-bj-bun 37019

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1539  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-v 3479  df-un 3967  df-sn 4631

This theorem is referenced by: (None)