Theorem bj-snfromadj 37404

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 4331	. 2 ⊢ (∅ ∪ {𝑥}) = {𝑥}
2		0ex 5236	. . 3 ⊢ ∅ ∈ V
3		bj-adjg1 37403	. . 3 ⊢ (∅ ∈ V → (∅ ∪ {𝑥}) ∈ V)
4	2, 3	ax-mp 5	. 2 ⊢ (∅ ∪ {𝑥}) ∈ V
5	1, 4	eqeltrri 2837	1 ⊢ {𝑥} ∈ V

Syntax hints:   ∈ wcel 2119  Vcvv 3432   ∪ cun 3888  ∅c0 4268  {csn 4562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-12 2189  ax-ext 2712  ax-nul 5235  ax-bj-adj 37402

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-v 3434  df-dif 3893  df-un 3895  df-nul 4269  df-sn 4563

This theorem is referenced by:  bj-prfromadj  37405