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Theorem iinuni 4174
 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 2854 . . . 4
2 elun 3488 . . . . 5
32ralbii 2729 . . . 4
4 vex 2959 . . . . . 6
54elint2 4057 . . . . 5
65orbi2i 506 . . . 4
71, 3, 63bitr4ri 270 . . 3
87abbii 2548 . 2
9 df-un 3325 . 2
10 df-iin 4096 . 2
118, 9, 103eqtr4i 2466 1
 Colors of variables: wff set class Syntax hints:   wo 358   wceq 1652   wcel 1725  cab 2422  wral 2705   cun 3318  cint 4050  ciin 4094 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 2417 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 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-v 2958  df-un 3325  df-int 4051  df-iin 4096
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