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Theorem intss 3947
 Description: Intersection of subclasses. (Contributed by NM, 14-Oct-1999.)
Assertion
Ref Expression
intss (A BB A)

Proof of Theorem intss
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 imim1 70 . . . . 5 ((y Ay B) → ((y Bx y) → (y Ax y)))
21al2imi 1561 . . . 4 (y(y Ay B) → (y(y Bx y) → y(y Ax y)))
3 vex 2862 . . . . 5 x V
43elint 3932 . . . 4 (x By(y Bx y))
53elint 3932 . . . 4 (x Ay(y Ax y))
62, 4, 53imtr4g 261 . . 3 (y(y Ay B) → (x Bx A))
76alrimiv 1631 . 2 (y(y Ay B) → x(x Bx A))
8 dfss2 3262 . 2 (A By(y Ay B))
9 dfss2 3262 . 2 (B Ax(x Bx A))
107, 8, 93imtr4i 257 1 (A BB A)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1540   ∈ wcel 1710   ⊆ wss 3257  ∩cint 3926 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-ss 3259  df-int 3927 This theorem is referenced by:  uniintsn  3963
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