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Theorem iundif2 4336
 Description: Indexed union of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use intiin 4323 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 3400 . . . . 5
21rexbii 2881 . . . 4
3 r19.42v 2931 . . . 4
4 rexnal 2836 . . . . . 6
5 vex 3034 . . . . . . 7
6 eliin 4275 . . . . . . 7
75, 6ax-mp 5 . . . . . 6
84, 7xchbinxr 318 . . . . 5
98anbi2i 708 . . . 4
102, 3, 93bitri 279 . . 3
11 eliun 4274 . . 3
12 eldif 3400 . . 3
1310, 11, 123bitr4i 285 . 2
1413eqriv 2468 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wb 189   wa 376   wceq 1452   wcel 1904  wral 2756  wrex 2757  cvv 3031   cdif 3387  ciun 4269  ciin 4270 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451 This theorem depends on definitions:  df-bi 190  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-rex 2762  df-v 3033  df-dif 3393  df-iun 4271  df-iin 4272 This theorem is referenced by:  iuncld  20137  pnrmopn  20436  alexsublem  21137  bcth3  22377  iundifdifd  28254  iundifdif  28255
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