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Theorem iindif2 4035
 Description: Indexed intersection of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use uniiun 4019 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 3645 . . . 4
2 eldif 3221 . . . . . 6
32bicomi 193 . . . . 5
43ralbii 2638 . . . 4
5 ralnex 2624 . . . . . 6
6 eliun 3973 . . . . . 6
75, 6xchbinxr 302 . . . . 5
87anbi2i 675 . . . 4
91, 4, 83bitr3g 278 . . 3
10 vex 2862 . . . 4
11 eliin 3974 . . . 4
1210, 11ax-mp 5 . . 3
13 eldif 3221 . . 3
149, 12, 133bitr4g 279 . 2
1514eqrdv 2351 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 176   wa 358   wceq 1642   wcel 1710   wne 2516  wral 2614  wrex 2615  cvv 2859   cdif 3206  c0 3550  ciun 3969  ciin 3970 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 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-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-nul 3551  df-iun 3971  df-iin 3972 This theorem is referenced by: (None)
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