Theorem bj-snfromadj 36979

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 4376	. 2 ⊢ (∅ ∪ {𝑥}) = {𝑥}
2		0ex 5287	. . 3 ⊢ ∅ ∈ V
3		bj-adjg1 36978	. . 3 ⊢ (∅ ∈ V → (∅ ∪ {𝑥}) ∈ V)
4	2, 3	ax-mp 5	. 2 ⊢ (∅ ∪ {𝑥}) ∈ V
5	1, 4	eqeltrri 2830	1 ⊢ {𝑥} ∈ V

Syntax hints:   ∈ wcel 2107  Vcvv 3463   ∪ cun 3929  ∅c0 4313  {csn 4606

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-12 2176  ax-ext 2706  ax-nul 5286  ax-bj-adj 36977

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-v 3465  df-dif 3934  df-un 3936  df-nul 4314  df-sn 4607

This theorem is referenced by:  bj-prfromadj  36980