Theorem bj-adjfrombun 38174

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 3458	. 2 ⊢ 𝑥 ∈ V
2		bj-snexg 38162	. . 3 ⊢ (𝑦 ∈ V → {𝑦} ∈ V)
3	2	elv 3459	. 2 ⊢ {𝑦} ∈ V
4		bj-unexg 38166	. 2 ⊢ ((𝑥 ∈ V ∧ {𝑦} ∈ V) → (𝑥 ∪ {𝑦}) ∈ V)
5	1, 3, 4	mp2an 705	1 ⊢ (𝑥 ∪ {𝑦}) ∈ V

This proof depends on syntax axioms:   ∈ wcel 2147  Vcvv 3454   ∪ cun 4152  {csn 4588

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-12 2216  ax-ext 2735  ax-bj-sn 38161  ax-bj-bun 38165

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-un 4154  df-sn 4589

This theorem is used by: (None)