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Theorem iindif2 3869
 Description: Indexed intersection of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use uniiun 3853 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
StepHypRef Expression
1 r19.28zv 3455 . . . 4
2 eldif 3088 . . . . . 6
32bicomi 195 . . . . 5
43ralbii 2531 . . . 4
5 ralnex 2517 . . . . . 6
6 eliun 3807 . . . . . 6
75, 6xchbinxr 304 . . . . 5
87anbi2i 678 . . . 4
91, 4, 83bitr3g 280 . . 3
10 vex 2730 . . . 4
11 eliin 3808 . . . 4
1210, 11ax-mp 10 . . 3
13 eldif 3088 . . 3
149, 12, 133bitr4g 281 . 2
1514eqrdv 2251 1
 Colors of variables: wff set class Syntax hints:   wn 5   wi 6   wb 178   wa 360   wceq 1619   wcel 1621   wne 2412  wral 2509  wrex 2510  cvv 2727   cdif 3075  c0 3362  ciun 3803  ciin 3804 This theorem is referenced by:  iincld  16608  clsval2  16619  mretopd  16661  hauscmplem  16965  cmpfi  16967 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-v 2729  df-dif 3081  df-nul 3363  df-iun 3805  df-iin 3806
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