Theorem bj-snfromadj 37367

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 4337	. 2 ⊢ (∅ ∪ {𝑥}) = {𝑥}
2		0ex 5242	. . 3 ⊢ ∅ ∈ V
3		bj-adjg1 37366	. . 3 ⊢ (∅ ∈ V → (∅ ∪ {𝑥}) ∈ V)
4	2, 3	ax-mp 5	. 2 ⊢ (∅ ∪ {𝑥}) ∈ V
5	1, 4	eqeltrri 2834	1 ⊢ {𝑥} ∈ V

Syntax hints:   ∈ wcel 2114  Vcvv 3430   ∪ cun 3888  ∅c0 4274  {csn 4568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-12 2185  ax-ext 2709  ax-nul 5241  ax-bj-adj 37365

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3432  df-dif 3893  df-un 3895  df-nul 4275  df-sn 4569

This theorem is referenced by:  bj-prfromadj  37368