Theorem bj-snfromadj 37029

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 4367	. 2 ⊢ (∅ ∪ {𝑥}) = {𝑥}
2		0ex 5270	. . 3 ⊢ ∅ ∈ V
3		bj-adjg1 37028	. . 3 ⊢ (∅ ∈ V → (∅ ∪ {𝑥}) ∈ V)
4	2, 3	ax-mp 5	. 2 ⊢ (∅ ∪ {𝑥}) ∈ V
5	1, 4	eqeltrri 2826	1 ⊢ {𝑥} ∈ V

Syntax hints:   ∈ wcel 2109  Vcvv 3455   ∪ cun 3920  ∅c0 4304  {csn 4597

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-12 2178  ax-ext 2702  ax-nul 5269  ax-bj-adj 37027

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3457  df-dif 3925  df-un 3927  df-nul 4305  df-sn 4598

This theorem is referenced by:  bj-prfromadj  37030