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Theorem iindif2 4399
 Description: Indexed intersection of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use uniiun 4383 to recover Enderton's theorem. (Contributed by NM, 5-Oct-2006.)
Assertion
Ref Expression
iindif2
Distinct variable groups:   ,   ,

Proof of Theorem iindif2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 r19.28zv 3924 . . . 4
2 eldif 3485 . . . . . 6
32bicomi 202 . . . . 5
43ralbii 2888 . . . 4
5 ralnex 2903 . . . . . 6
6 eliun 4335 . . . . . 6
75, 6xchbinxr 311 . . . . 5
87anbi2i 694 . . . 4
91, 4, 83bitr3g 287 . . 3
10 vex 3112 . . . 4
11 eliin 4336 . . . 4
1210, 11ax-mp 5 . . 3
13 eldif 3485 . . 3
149, 12, 133bitr4g 288 . 2
1514eqrdv 2454 1
 Colors of variables: wff setvar class Syntax hints:  -.wn 3  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  e.wcel 1818  =/=wne 2652  A.wral 2807  E.wrex 2808   cvv 3109  \cdif 3472   c0 3784  U_ciun 4330  |^|_ciin 4331 This theorem is referenced by:  iinvdif  4402  iincld  19540  clsval2  19551  mretopd  19593  hauscmplem  19906  cmpfi  19908 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-v 3111  df-dif 3478  df-nul 3785  df-iun 4332  df-iin 4333
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