Theorem bj-adjfrombun 37399

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 3435	. 2 ⊢ 𝑥 ∈ V
2		bj-snexg 37387	. . 3 ⊢ (𝑦 ∈ V → {𝑦} ∈ V)
3	2	elv 3436	. 2 ⊢ {𝑦} ∈ V
4		bj-unexg 37391	. 2 ⊢ ((𝑥 ∈ V ∧ {𝑦} ∈ V) → (𝑥 ∪ {𝑦}) ∈ V)
5	1, 3, 4	mp2an 698	1 ⊢ (𝑥 ∪ {𝑦}) ∈ V

Syntax hints:   ∈ wcel 2119  Vcvv 3431   ∪ cun 3881  {csn 4555

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-12 2189  ax-ext 2711  ax-bj-sn 37386  ax-bj-bun 37390

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-v 3433  df-un 3888  df-sn 4556

This theorem is referenced by: (None)