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Theorem iinuni 3985
 Description: A relationship involving union and indexed intersection. Exercise 23 of [Enderton] p. 33. (Contributed by NM, 25-Nov-2003.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
Assertion
Ref Expression
iinuni
Distinct variable groups:   ,   ,

Proof of Theorem iinuni
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 r19.32v 2686 . . . 4
2 elun 3316 . . . . 5
32ralbii 2567 . . . 4
4 vex 2791 . . . . . 6
54elint2 3869 . . . . 5
65orbi2i 505 . . . 4
71, 3, 63bitr4ri 269 . . 3
87abbii 2395 . 2
9 df-un 3157 . 2
10 df-iin 3908 . 2
118, 9, 103eqtr4i 2313 1
 Colors of variables: wff set class Syntax hints:   wo 357   wceq 1623   wcel 1684  cab 2269  wral 2543   cun 3150  cint 3862  ciin 3906 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-v 2790  df-un 3157  df-int 3863  df-iin 3908
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