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Theorem int0 3620
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 3222 . . . . . 6 ¬ y
21pm2.21i 574 . . . . 5 (y ∅ → x y)
32ax-gen 1335 . . . 4 y(y ∅ → x y)
4 equid 1586 . . . 4 x = x
53, 42th 163 . . 3 (y(y ∅ → x y) ↔ x = x)
65abbii 2150 . 2 {xy(y ∅ → x y)} = {xx = x}
7 df-int 3607 . 2 ∅ = {xy(y ∅ → x y)}
8 df-v 2553 . 2 V = {xx = x}
96, 7, 83eqtr4i 2067 1 ∅ = V
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1240   = wceq 1242   wcel 1390  {cab 2023  Vcvv 2551  c0 3218   cint 3606
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 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-dif 2914  df-nul 3219  df-int 3607
This theorem is referenced by:  rint0  3645  intexr  3895  bj-intexr  9339
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