MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  intpr Structured version   Visualization version   GIF version

Theorem intpr 4985
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 4984. (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 4984 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} = (𝐴𝐵))
41, 2, 3mp2an 690 1 {𝐴, 𝐵} = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  wcel 2106  Vcvv 3474  cin 3946  {cpr 4629   cint 4949
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-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-un 3952  df-in 3954  df-sn 4628  df-pr 4630  df-int 4950
This theorem is referenced by:  intprgOLD  4987  uniintsn  4990  op1stb  5470  fiint  9320  shincli  30602  chincli  30700
  Copyright terms: Public domain W3C validator