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Theorem iinun2 4149
 Description: Indexed intersection of union. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 4137 to recover Enderton's theorem. (Contributed by NM, 19-Aug-2004.)
Assertion
Ref Expression
iinun2
Distinct variable group:   ,
Allowed substitution hints:   ()   ()

Proof of Theorem iinun2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 r19.32v 2846 . . . 4
2 elun 3480 . . . . 5
32ralbii 2721 . . . 4
4 vex 2951 . . . . . 6
5 eliin 4090 . . . . . 6
64, 5ax-mp 8 . . . . 5
76orbi2i 506 . . . 4
81, 3, 73bitr4i 269 . . 3
9 eliin 4090 . . . 4
104, 9ax-mp 8 . . 3
11 elun 3480 . . 3
128, 10, 113bitr4i 269 . 2
1312eqriv 2432 1
 Colors of variables: wff set class Syntax hints:   wb 177   wo 358   wceq 1652   wcel 1725  wral 2697  cvv 2948   cun 3310  ciin 4086 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-v 2950  df-un 3317  df-iin 4088
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