Theorem bj-snfromadj 37661

Description: Singleton from adjunction and empty set. (Contributed by BJ, 19-Jan-2025.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-snfromadj	⊢ {𝑥} ∈ V

Proof of Theorem bj-snfromadj

Step	Hyp	Ref	Expression
1		0un 4354	. 2 ⊢ (∅ ∪ {𝑥}) = {𝑥}
2		0ex 5271	. . 3 ⊢ ∅ ∈ V
3		bj-adjg1 37660	. . 3 ⊢ (∅ ∈ V → (∅ ∪ {𝑥}) ∈ V)
4	2, 3	ax-mp 5	. 2 ⊢ (∅ ∪ {𝑥}) ∈ V
5	1, 4	eqeltrri 2860	1 ⊢ {𝑥} ∈ V

Syntax hints:   ∈ wcel 2143  Vcvv 3455   ∪ cun 3904  ∅c0 4287  {csn 4590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-12 2213  ax-ext 2735  ax-nul 5270  ax-bj-adj 37659

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-dif 3909  df-un 3911  df-nul 4288  df-sn 4591

This theorem is referenced by:  bj-prfromadj  37662