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Theorem bj-prfromadj 35730
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
StepHypRef Expression
1 df-pr 4625 . 2 {𝑥, 𝑦} = ({𝑥} ∪ {𝑦})
2 bj-snfromadj 35729 . . 3 {𝑥} ∈ V
3 bj-adjg1 35728 . . 3 ({𝑥} ∈ V → ({𝑥} ∪ {𝑦}) ∈ V)
42, 3ax-mp 5 . 2 ({𝑥} ∪ {𝑦}) ∈ V
51, 4eqeltri 2828 1 {𝑥, 𝑦} ∈ V
Colors of variables: wff setvar class
Syntax hints:  wcel 2106  Vcvv 3473  cun 3942  {csn 4622  {cpr 4624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-12 2171  ax-ext 2702  ax-nul 5299  ax-bj-adj 35727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-dif 3947  df-un 3949  df-nul 4319  df-sn 4623  df-pr 4625
This theorem is referenced by: (None)
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