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Theorem intss1 3941
 Description: An element of a class includes the intersection of the class. Exercise 4 of [TakeutiZaring] p. 44 (with correction), generalized to classes. (Contributed by NM, 18-Nov-1995.)
Assertion
Ref Expression
intss1 (A BB A)

Proof of Theorem intss1
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2862 . . . 4 x V
21elint 3932 . . 3 (x By(y Bx y))
3 eleq1 2413 . . . . . 6 (y = A → (y BA B))
4 eleq2 2414 . . . . . 6 (y = A → (x yx A))
53, 4imbi12d 311 . . . . 5 (y = A → ((y Bx y) ↔ (A Bx A)))
65spcgv 2939 . . . 4 (A B → (y(y Bx y) → (A Bx A)))
76pm2.43a 45 . . 3 (A B → (y(y Bx y) → x A))
82, 7syl5bi 208 . 2 (A B → (x Bx A))
98ssrdv 3278 1 (A BB A)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1540   = wceq 1642   ∈ wcel 1710   ⊆ wss 3257  ∩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-ss 3259  df-int 3927 This theorem is referenced by:  intminss  3952  intmin3  3954  intab  3956  int0el  3957  peano5  4409  spfininduct  4540  clos1induct  5880
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