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Theorem intprg 4710
Description: The intersection of a pair is the intersection of its members. Closed form of intpr 4709. Theorem 71 of [Suppes] p. 42. (Contributed by FL, 27-Apr-2008.)
Assertion
Ref Expression
intprg ((𝐴𝑉𝐵𝑊) → {𝐴, 𝐵} = (𝐴𝐵))

Proof of Theorem intprg
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 preq1 4466 . . . 4 (𝑥 = 𝐴 → {𝑥, 𝑦} = {𝐴, 𝑦})
21inteqd 4681 . . 3 (𝑥 = 𝐴 {𝑥, 𝑦} = {𝐴, 𝑦})
3 ineq1 4013 . . 3 (𝑥 = 𝐴 → (𝑥𝑦) = (𝐴𝑦))
42, 3eqeq12d 2828 . 2 (𝑥 = 𝐴 → ( {𝑥, 𝑦} = (𝑥𝑦) ↔ {𝐴, 𝑦} = (𝐴𝑦)))
5 preq2 4467 . . . 4 (𝑦 = 𝐵 → {𝐴, 𝑦} = {𝐴, 𝐵})
65inteqd 4681 . . 3 (𝑦 = 𝐵 {𝐴, 𝑦} = {𝐴, 𝐵})
7 ineq2 4014 . . 3 (𝑦 = 𝐵 → (𝐴𝑦) = (𝐴𝐵))
86, 7eqeq12d 2828 . 2 (𝑦 = 𝐵 → ( {𝐴, 𝑦} = (𝐴𝑦) ↔ {𝐴, 𝐵} = (𝐴𝐵)))
9 vex 3401 . . 3 𝑥 ∈ V
10 vex 3401 . . 3 𝑦 ∈ V
119, 10intpr 4709 . 2 {𝑥, 𝑦} = (𝑥𝑦)
124, 8, 11vtocl2g 3470 1 ((𝐴𝑉𝐵𝑊) → {𝐴, 𝐵} = (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1637  wcel 2157  cin 3775  {cpr 4379   cint 4676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ral 3108  df-v 3400  df-un 3781  df-in 3783  df-sn 4378  df-pr 4380  df-int 4677
This theorem is referenced by:  intsng  4711  inelfi  8566  mreincl  16467  subrgin  19010  lssincl  19175  incld  21065  difelsiga  30527  inelpisys  30548  inidl  34142
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