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Theorem intpr 3638
Description: The intersection of a pair is the intersection of its members. Theorem 71 of [Suppes] p. 42. (Contributed by NM, 14-Oct-1999.)
Hypotheses
Ref Expression
intpr.1 A V
intpr.2 B V
Assertion
Ref Expression
intpr {A, B} = (AB)

Proof of Theorem intpr
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 19.26 1367 . . . 4 (y((y = Ax y) (y = Bx y)) ↔ (y(y = Ax y) y(y = Bx y)))
2 vex 2554 . . . . . . . 8 y V
32elpr 3385 . . . . . . 7 (y {A, B} ↔ (y = A y = B))
43imbi1i 227 . . . . . 6 ((y {A, B} → x y) ↔ ((y = A y = B) → x y))
5 jaob 630 . . . . . 6 (((y = A y = B) → x y) ↔ ((y = Ax y) (y = Bx y)))
64, 5bitri 173 . . . . 5 ((y {A, B} → x y) ↔ ((y = Ax y) (y = Bx y)))
76albii 1356 . . . 4 (y(y {A, B} → x y) ↔ y((y = Ax y) (y = Bx y)))
8 intpr.1 . . . . . 6 A V
98clel4 2674 . . . . 5 (x Ay(y = Ax y))
10 intpr.2 . . . . . 6 B V
1110clel4 2674 . . . . 5 (x By(y = Bx y))
129, 11anbi12i 433 . . . 4 ((x A x B) ↔ (y(y = Ax y) y(y = Bx y)))
131, 7, 123bitr4i 201 . . 3 (y(y {A, B} → x y) ↔ (x A x B))
14 vex 2554 . . . 4 x V
1514elint 3612 . . 3 (x {A, B} ↔ y(y {A, B} → x y))
16 elin 3120 . . 3 (x (AB) ↔ (x A x B))
1713, 15, 163bitr4i 201 . 2 (x {A, B} ↔ x (AB))
1817eqriv 2034 1 {A, B} = (AB)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wo 628  wal 1240   = wceq 1242   wcel 1390  Vcvv 2551  cin 2910  {cpr 3368   cint 3606
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-sn 3373  df-pr 3374  df-int 3607
This theorem is referenced by:  intprg  3639  op1stb  4175
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