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Theorem iundif2 4150
 Description: Indexed union of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use intiin 4137 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 3322 . . . . 5
21rexbii 2722 . . . 4
3 r19.42v 2854 . . . 4
4 rexnal 2708 . . . . . 6
5 vex 2951 . . . . . . 7
6 eliin 4090 . . . . . . 7
75, 6ax-mp 8 . . . . . 6
84, 7xchbinxr 303 . . . . 5
98anbi2i 676 . . . 4
102, 3, 93bitri 263 . . 3
11 eliun 4089 . . 3
12 eldif 3322 . . 3
1310, 11, 123bitr4i 269 . 2
1413eqriv 2432 1
 Colors of variables: wff set class Syntax hints:   wn 3   wb 177   wa 359   wceq 1652   wcel 1725  wral 2697  wrex 2698  cvv 2948   cdif 3309  ciun 4085  ciin 4086 This theorem is referenced by:  iuncld  17101  pnrmopn  17399  alexsublem  18067  bcth3  19276  iundifdifd  24004  iundifdif  24005 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-ral 2702  df-rex 2703  df-v 2950  df-dif 3315  df-iun 4087  df-iin 4088
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