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Theorem ninexg 4097
 Description: The anti-intersection of two sets is a set. (Contributed by SF, 12-Jan-2015.)
Assertion
Ref Expression
ninexg &ncap

Proof of Theorem ninexg
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nineq1 3234 . . 3 &ncap &ncap
21eleq1d 2419 . 2 &ncap &ncap
3 nineq2 3235 . . 3 &ncap &ncap
43eleq1d 2419 . 2 &ncap &ncap
5 ax-nin 4078 . . 3
6 isset 2863 . . . 4 &ncap &ncap
7 dfcleq 2347 . . . . . 6 &ncap &ncap
8 vex 2862 . . . . . . . . 9
98elnin 3224 . . . . . . . 8 &ncap
109bibi2i 304 . . . . . . 7 &ncap
1110albii 1566 . . . . . 6 &ncap
127, 11bitri 240 . . . . 5 &ncap
1312exbii 1582 . . . 4 &ncap
146, 13bitri 240 . . 3 &ncap
155, 14mpbir 200 . 2 &ncap
162, 4, 15vtocl2g 2918 1 &ncap
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 176   wa 358   wnan 1287  wal 1540  wex 1541   wceq 1642   wcel 1710  cvv 2859   &ncap cnin 3204 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  ax-nin 4078 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 This theorem is referenced by:  ninex  4098  complexg  4099  inexg  4100  unexg  4101
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