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Theorem intprg 3774
Description: The intersection of a pair is the intersection of its members. Closed form of intpr 3773. 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 3570 . . . 4 (𝑥 = 𝐴 → {𝑥, 𝑦} = {𝐴, 𝑦})
21inteqd 3746 . . 3 (𝑥 = 𝐴 {𝑥, 𝑦} = {𝐴, 𝑦})
3 ineq1 3240 . . 3 (𝑥 = 𝐴 → (𝑥𝑦) = (𝐴𝑦))
42, 3eqeq12d 2132 . 2 (𝑥 = 𝐴 → ( {𝑥, 𝑦} = (𝑥𝑦) ↔ {𝐴, 𝑦} = (𝐴𝑦)))
5 preq2 3571 . . . 4 (𝑦 = 𝐵 → {𝐴, 𝑦} = {𝐴, 𝐵})
65inteqd 3746 . . 3 (𝑦 = 𝐵 {𝐴, 𝑦} = {𝐴, 𝐵})
7 ineq2 3241 . . 3 (𝑦 = 𝐵 → (𝐴𝑦) = (𝐴𝐵))
86, 7eqeq12d 2132 . 2 (𝑦 = 𝐵 → ( {𝐴, 𝑦} = (𝐴𝑦) ↔ {𝐴, 𝐵} = (𝐴𝐵)))
9 vex 2663 . . 3 𝑥 ∈ V
10 vex 2663 . . 3 𝑦 ∈ V
119, 10intpr 3773 . 2 {𝑥, 𝑦} = (𝑥𝑦)
124, 8, 11vtocl2g 2724 1 ((𝐴𝑉𝐵𝑊) → {𝐴, 𝐵} = (𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1316  wcel 1465  cin 3040  {cpr 3498   cint 3741
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-v 2662  df-un 3045  df-in 3047  df-sn 3503  df-pr 3504  df-int 3742
This theorem is referenced by:  intsng  3775  op1stbg  4370
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