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Theorem iindif2 4347
 Description: Indexed intersection of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use uniiun 4331 to recover Enderton's theorem. (Contributed by NM, 5-Oct-2006.)
Assertion
Ref Expression
iindif2
Distinct variable groups:   ,   ,
Allowed substitution hint:   ()

Proof of Theorem iindif2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 r19.28zv 3864 . . . 4
2 eldif 3414 . . . . . 6
32bicomi 206 . . . . 5
43ralbii 2819 . . . 4
5 ralnex 2834 . . . . . 6
6 eliun 4283 . . . . . 6
75, 6xchbinxr 313 . . . . 5
87anbi2i 700 . . . 4
91, 4, 83bitr3g 291 . . 3
10 vex 3048 . . . 4
11 eliin 4284 . . . 4
1210, 11ax-mp 5 . . 3
13 eldif 3414 . . 3
149, 12, 133bitr4g 292 . 2
1514eqrdv 2449 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 188   wa 371   wceq 1444   wcel 1887   wne 2622  wral 2737  wrex 2738  cvv 3045   cdif 3401  c0 3731  ciun 4278  ciin 4279 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431 This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-v 3047  df-dif 3407  df-nul 3732  df-iun 4280  df-iin 4281 This theorem is referenced by:  iinvdif  4350  iincld  20054  clsval2  20065  mretopd  20108  hauscmplem  20421  cmpfi  20423  sigapildsyslem  28983  saliincl  38186
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