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Theorem iindif2 4152
 Description: Indexed intersection of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use uniiun 4136 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 3715 . . . 4
2 eldif 3322 . . . . . 6
32bicomi 194 . . . . 5
43ralbii 2721 . . . 4
5 ralnex 2707 . . . . . 6
6 eliun 4089 . . . . . 6
75, 6xchbinxr 303 . . . . 5
87anbi2i 676 . . . 4
91, 4, 83bitr3g 279 . . 3
10 vex 2951 . . . 4
11 eliin 4090 . . . 4
1210, 11ax-mp 8 . . 3
13 eldif 3322 . . 3
149, 12, 133bitr4g 280 . 2
1514eqrdv 2433 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 177   wa 359   wceq 1652   wcel 1725   wne 2598  wral 2697  wrex 2698  cvv 2948   cdif 3309  c0 3620  ciun 4085  ciin 4086 This theorem is referenced by:  iincld  17095  clsval2  17106  mretopd  17148  hauscmplem  17461  cmpfi  17463 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-v 2950  df-dif 3315  df-nul 3621  df-iun 4087  df-iin 4088
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