Theorem bj-prfromadj 37535

Description: Unordered pair from adjunction. (Contributed by BJ, 19-Jan-2025.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-prfromadj	⊢ {𝑥, 𝑦} ∈ V

Proof of Theorem bj-prfromadj

Step	Hyp	Ref	Expression
1		df-pr 4587	. 2 ⊢ {𝑥, 𝑦} = ({𝑥} ∪ {𝑦})
2		bj-snfromadj 37534	. . 3 ⊢ {𝑥} ∈ V
3		bj-adjg1 37533	. . 3 ⊢ ({𝑥} ∈ V → ({𝑥} ∪ {𝑦}) ∈ V)
4	2, 3	ax-mp 5	. 2 ⊢ ({𝑥} ∪ {𝑦}) ∈ V
5	1, 4	eqeltri 2860	1 ⊢ {𝑥, 𝑦} ∈ V

Syntax hints:   ∈ wcel 2144  Vcvv 3456   ∪ cun 3904  {csn 4584  {cpr 4586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-12 2214  ax-ext 2736  ax-nul 5258  ax-bj-adj 37532

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-v 3458  df-dif 3909  df-un 3911  df-nul 4288  df-sn 4585  df-pr 4587

This theorem is referenced by: (None)