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Theorem int0 3599
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 3201 . . . . . 6 ¬ y
21pm2.21i 562 . . . . 5 (y ∅ → x y)
32ax-gen 1314 . . . 4 y(y ∅ → x y)
4 equid 1567 . . . 4 x = x
53, 42th 163 . . 3 (y(y ∅ → x y) ↔ x = x)
65abbii 2131 . 2 {xy(y ∅ → x y)} = {xx = x}
7 df-int 3586 . 2 ∅ = {xy(y ∅ → x y)}
8 df-v 2533 . 2 V = {xx = x}
96, 7, 83eqtr4i 2048 1 ∅ = V
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1224   = wceq 1226   wcel 1370  {cab 2004  Vcvv 2531  c0 3197   cint 3585
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-in1 532  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-v 2533  df-dif 2893  df-nul 3198  df-int 3586
This theorem is referenced by:  rint0  3624  intexr  3874  bj-intexr  7270
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