Theorem bj-snfromadj 37109

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 4345	. 2 ⊢ (∅ ∪ {𝑥}) = {𝑥}
2		0ex 5247	. . 3 ⊢ ∅ ∈ V
3		bj-adjg1 37108	. . 3 ⊢ (∅ ∈ V → (∅ ∪ {𝑥}) ∈ V)
4	2, 3	ax-mp 5	. 2 ⊢ (∅ ∪ {𝑥}) ∈ V
5	1, 4	eqeltrri 2830	1 ⊢ {𝑥} ∈ V

Syntax hints:   ∈ wcel 2113  Vcvv 3437   ∪ cun 3896  ∅c0 4282  {csn 4575

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-12 2182  ax-ext 2705  ax-nul 5246  ax-bj-adj 37107

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-v 3439  df-dif 3901  df-un 3903  df-nul 4283  df-sn 4576

This theorem is referenced by:  bj-prfromadj  37110