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Theorem iundif2 4033
 Description: Indexed union of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use intiin 4020 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 3221 . . . . 5
21rexbii 2639 . . . 4
3 r19.42v 2765 . . . 4
4 rexnal 2625 . . . . . 6
5 vex 2862 . . . . . . 7
6 eliin 3974 . . . . . . 7
75, 6ax-mp 8 . . . . . 6
84, 7xchbinxr 302 . . . . 5
98anbi2i 675 . . . 4
102, 3, 93bitri 262 . . 3
11 eliun 3973 . . 3
12 eldif 3221 . . 3
1310, 11, 123bitr4i 268 . 2
1413eqriv 2350 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wb 176   wa 358   wceq 1642   wcel 1710  wral 2614  wrex 2615  cvv 2859   cdif 3206  ciun 3969  ciin 3970 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-iun 3971  df-iin 3972 This theorem is referenced by: (None)
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