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Theorem iundif2 3969
 Description: Indexed union of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use intiin 3956 to recover Enderton's theorem. (Contributed by NM, 19-Aug-2004.)
Assertion
Ref Expression
iundif2
Distinct variable group:   ,
Allowed substitution hints:   ()   ()

Proof of Theorem iundif2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eldif 3162 . . . . 5
21rexbii 2568 . . . 4
3 r19.42v 2694 . . . 4
4 rexnal 2554 . . . . . 6
5 vex 2791 . . . . . . 7
6 eliin 3910 . . . . . . 7
75, 6ax-mp 8 . . . . . 6
84, 7xchbinxr 302 . . . . 5
98anbi2i 675 . . . 4
102, 3, 93bitri 262 . . 3
11 eliun 3909 . . 3
12 eldif 3162 . . 3
1310, 11, 123bitr4i 268 . 2
1413eqriv 2280 1
 Colors of variables: wff set class Syntax hints:   wn 3   wb 176   wa 358   wceq 1623   wcel 1684  wral 2543  wrex 2544  cvv 2788   cdif 3149  ciun 3905  ciin 3906 This theorem is referenced by:  iuncld  16782  pnrmopn  17071  alexsublem  17738  bcth3  18753  iundifdifd  23159  iundifdif  23160 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-v 2790  df-dif 3155  df-iun 3907  df-iin 3908
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