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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 ((A V B W) → (AB) V)

Proof of Theorem ninexg
Dummy variables x y z w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nineq1 3234 . . 3 (x = A → (xy) = (Ay))
21eleq1d 2419 . 2 (x = A → ((xy) V ↔ (Ay) V))
3 nineq2 3235 . . 3 (y = B → (Ay) = (AB))
43eleq1d 2419 . 2 (y = B → ((Ay) V ↔ (AB) V))
5 ax-nin 4078 . . 3 zw(w z ↔ (w x w y))
6 isset 2863 . . . 4 ((xy) V ↔ z z = (xy))
7 dfcleq 2347 . . . . . 6 (z = (xy) ↔ w(w zw (xy)))
8 vex 2862 . . . . . . . . 9 w V
98elnin 3224 . . . . . . . 8 (w (xy) ↔ (w x w y))
109bibi2i 304 . . . . . . 7 ((w zw (xy)) ↔ (w z ↔ (w x w y)))
1110albii 1566 . . . . . 6 (w(w zw (xy)) ↔ w(w z ↔ (w x w y)))
127, 11bitri 240 . . . . 5 (z = (xy) ↔ w(w z ↔ (w x w y)))
1312exbii 1582 . . . 4 (z z = (xy) ↔ zw(w z ↔ (w x w y)))
146, 13bitri 240 . . 3 ((xy) V ↔ zw(w z ↔ (w x w y)))
155, 14mpbir 200 . 2 (xy) V
162, 4, 15vtocl2g 2918 1 ((A V B W) → (AB) V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ⊼ wnan 1287  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2859   ⩃ cnin 3204 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  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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