Theorem bj-snfromadj 36446

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 4388	. 2 ⊢ (∅ ∪ {𝑥}) = {𝑥}
2		0ex 5301	. . 3 ⊢ ∅ ∈ V
3		bj-adjg1 36445	. . 3 ⊢ (∅ ∈ V → (∅ ∪ {𝑥}) ∈ V)
4	2, 3	ax-mp 5	. 2 ⊢ (∅ ∪ {𝑥}) ∈ V
5	1, 4	eqeltrri 2825	1 ⊢ {𝑥} ∈ V

Syntax hints:   ∈ wcel 2099  Vcvv 3469   ∪ cun 3942  ∅c0 4318  {csn 4624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-12 2164  ax-ext 2698  ax-nul 5300  ax-bj-adj 36444

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-v 3471  df-dif 3947  df-un 3949  df-nul 4319  df-sn 4625

This theorem is referenced by:  bj-prfromadj  36447