Theorem bj-snfromadj 37039

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 4403	. 2 ⊢ (∅ ∪ {𝑥}) = {𝑥}
2		0ex 5314	. . 3 ⊢ ∅ ∈ V
3		bj-adjg1 37038	. . 3 ⊢ (∅ ∈ V → (∅ ∪ {𝑥}) ∈ V)
4	2, 3	ax-mp 5	. 2 ⊢ (∅ ∪ {𝑥}) ∈ V
5	1, 4	eqeltrri 2837	1 ⊢ {𝑥} ∈ V

Syntax hints:   ∈ wcel 2107  Vcvv 3479   ∪ cun 3962  ∅c0 4340  {csn 4632

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-12 2176  ax-ext 2707  ax-nul 5313  ax-bj-adj 37037

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1541  df-fal 1551  df-ex 1778  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-v 3481  df-dif 3967  df-un 3969  df-nul 4341  df-sn 4633

This theorem is referenced by:  bj-prfromadj  37040