Theorem bj-adjfrombun 36656

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 3465	. 2 ⊢ 𝑥 ∈ V
2		bj-snexg 36644	. . 3 ⊢ (𝑦 ∈ V → {𝑦} ∈ V)
3	2	elv 3467	. 2 ⊢ {𝑦} ∈ V
4		bj-unexg 36648	. 2 ⊢ ((𝑥 ∈ V ∧ {𝑦} ∈ V) → (𝑥 ∪ {𝑦}) ∈ V)
5	1, 3, 4	mp2an 690	1 ⊢ (𝑥 ∪ {𝑦}) ∈ V

Syntax hints:   ∈ wcel 2098  Vcvv 3461   ∪ cun 3942  {csn 4630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-12 2166  ax-ext 2696  ax-bj-sn 36643  ax-bj-bun 36647

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-v 3463  df-un 3949  df-sn 4631

This theorem is referenced by: (None)