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Theorem intpr 3959
 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 1593 . . . 4 (y((y = Ax y) (y = Bx y)) ↔ (y(y = Ax y) y(y = Bx y)))
2 vex 2862 . . . . . . . 8 y V
32elpr 3751 . . . . . . 7 (y {A, B} ↔ (y = A y = B))
43imbi1i 315 . . . . . 6 ((y {A, B} → x y) ↔ ((y = A y = B) → x y))
5 jaob 758 . . . . . 6 (((y = A y = B) → x y) ↔ ((y = Ax y) (y = Bx y)))
64, 5bitri 240 . . . . 5 ((y {A, B} → x y) ↔ ((y = Ax y) (y = Bx y)))
76albii 1566 . . . 4 (y(y {A, B} → x y) ↔ y((y = Ax y) (y = Bx y)))
8 intpr.1 . . . . . 6 A V
98clel4 2978 . . . . 5 (x Ay(y = Ax y))
10 intpr.2 . . . . . 6 B V
1110clel4 2978 . . . . 5 (x By(y = Bx y))
129, 11anbi12i 678 . . . 4 ((x A x B) ↔ (y(y = Ax y) y(y = Bx y)))
131, 7, 123bitr4i 268 . . 3 (y(y {A, B} → x y) ↔ (x A x B))
14 vex 2862 . . . 4 x V
1514elint 3932 . . 3 (x {A, B} ↔ y(y {A, B} → x y))
16 elin 3219 . . 3 (x (AB) ↔ (x A x B))
1713, 15, 163bitr4i 268 . 2 (x {A, B} ↔ x (AB))
1817eqriv 2350 1 {A, B} = (AB)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 357   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710  Vcvv 2859   ∩ cin 3208  {cpr 3738  ∩cint 3926 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-sn 3741  df-pr 3742  df-int 3927 This theorem is referenced by:  intprg  3960  uniintsn  3963
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