Theorem bj-adjfrombun 37715

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 3461	. 2 ⊢ 𝑥 ∈ V
2		bj-snexg 37703	. . 3 ⊢ (𝑦 ∈ V → {𝑦} ∈ V)
3	2	elv 3462	. 2 ⊢ {𝑦} ∈ V
4		bj-unexg 37707	. 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 2146  Vcvv 3457   ∪ cun 3904  {csn 4591

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-12 2216  ax-ext 2737  ax-bj-sn 37702  ax-bj-bun 37706

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-un 3911  df-sn 4592

This theorem is used by: (None)