Theorem bj-snfromadj 36579

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 5302	. . 3 ⊢ ∅ ∈ V
3		bj-adjg1 36578	. . 3 ⊢ (∅ ∈ V → (∅ ∪ {𝑥}) ∈ V)
4	2, 3	ax-mp 5	. 2 ⊢ (∅ ∪ {𝑥}) ∈ V
5	1, 4	eqeltrri 2822	1 ⊢ {𝑥} ∈ V

Syntax hints:   ∈ wcel 2098  Vcvv 3463   ∪ cun 3938  ∅c0 4318  {csn 4624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-12 2166  ax-ext 2696  ax-nul 5301  ax-bj-adj 36577

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-v 3465  df-dif 3943  df-un 3945  df-nul 4319  df-sn 4625

This theorem is referenced by:  bj-prfromadj  36580