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Theorem int0 3940
 Description: The intersection of the empty set is the universal class. Exercise 2 of [TakeutiZaring] p. 44. (Contributed by NM, 18-Aug-1993.)
Assertion
Ref Expression
int0 = V

Proof of Theorem int0
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 noel 3554 . . . . . 6 ¬ y
21pm2.21i 123 . . . . 5 (y x y)
32ax-gen 1546 . . . 4 y(y x y)
4 eqid 2353 . . . 4 x = x
53, 42th 230 . . 3 (y(y x y) ↔ x = x)
65abbii 2465 . 2 {x y(y x y)} = {x x = x}
7 df-int 3927 . 2 = {x y(y x y)}
8 df-v 2861 . 2 V = {x x = x}
96, 7, 83eqtr4i 2383 1 = V
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1540   = wceq 1642   ∈ wcel 1710  {cab 2339  Vcvv 2859  ∅c0 3550  ∩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-dif 3215  df-nul 3551  df-int 3927 This theorem is referenced by:  unissint  3950  uniintsn  3963  rint0  3966
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