Theorem bj-prfromadj 38000

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 4586	. 2 ⊢ {𝑥, 𝑦} = ({𝑥} ∪ {𝑦})
2		bj-snfromadj 37999	. . 3 ⊢ {𝑥} ∈ V
3		bj-adjg1 37998	. . 3 ⊢ ({𝑥} ∈ V → ({𝑥} ∪ {𝑦}) ∈ V)
4	2, 3	ax-mp 5	. 2 ⊢ ({𝑥} ∪ {𝑦}) ∈ V
5	1, 4	eqeltri 2856	1 ⊢ {𝑥, 𝑦} ∈ V

This proof depends on syntax axioms:   ∈ wcel 2144  Vcvv 3450   ∪ cun 4149  {csn 4583  {cpr 4585

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-12 2212  ax-ext 2732  ax-nul 5259  ax-bj-adj 37997

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-dif 4047  df-un 4151  df-nul 4279  df-sn 4584  df-pr 4586

This theorem is used by: (None)