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Theorem intpr 3621
 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 1350 . . . 4 (y((y = Ax y) (y = Bx y)) ↔ (y(y = Ax y) y(y = Bx y)))
2 vex 2538 . . . . . . . 8 y V
32elpr 3368 . . . . . . 7 (y {A, B} ↔ (y = A y = B))
43imbi1i 227 . . . . . 6 ((y {A, B} → x y) ↔ ((y = A y = B) → x y))
5 jaob 618 . . . . . 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 1339 . . . 4 (y(y {A, B} → x y) ↔ y((y = Ax y) (y = Bx y)))
8 intpr.1 . . . . . 6 A V
98clel4 2657 . . . . 5 (x Ay(y = Ax y))
10 intpr.2 . . . . . 6 B V
1110clel4 2657 . . . . 5 (x By(y = Bx y))
129, 11anbi12i 436 . . . 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 2538 . . . 4 x V
1514elint 3595 . . 3 (x {A, B} ↔ y(y {A, B} → x y))
16 elin 3103 . . 3 (x (AB) ↔ (x A x B))
1713, 15, 163bitr4i 201 . 2 (x {A, B} ↔ x (AB))
1817eqriv 2019 1 {A, B} = (AB)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∨ wo 616  ∀wal 1226   = wceq 1228   ∈ wcel 1374  Vcvv 2535   ∩ cin 2893  {cpr 3351  ∩ cint 3589 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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-un 2899  df-in 2901  df-sn 3356  df-pr 3357  df-int 3590 This theorem is referenced by:  intprg  3622  op1stb  4159
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