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Theorem intpr 4986
Description: The intersection of a pair is the intersection of its members. Theorem 71 of [Suppes] p. 42. (Contributed by NM, 14-Oct-1999.) Prove from intprg 4985. (Revised by BJ, 1-Sep-2024.)
Hypotheses
Ref Expression
intpr.1 𝐴 ∈ V
intpr.2 𝐵 ∈ V
Assertion
Ref Expression
intpr {𝐴, 𝐵} = (𝐴𝐵)

Proof of Theorem intpr
StepHypRef Expression
1 intpr.1 . 2 𝐴 ∈ V
2 intpr.2 . 2 𝐵 ∈ V
3 intprg 4985 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} = (𝐴𝐵))
41, 2, 3mp2an 689 1 {𝐴, 𝐵} = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wcel 2105  Vcvv 3473  cin 3947  {cpr 4630   cint 4950
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-un 3953  df-in 3955  df-sn 4629  df-pr 4631  df-int 4951
This theorem is referenced by:  intprgOLD  4988  uniintsn  4991  op1stb  5471  fiint  9330  shincli  31048  chincli  31146
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