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Theorem iinuni 4358
 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 2922 . . . 4
2 elun 3565 . . . . 5
32ralbii 2823 . . . 4
4 vex 3034 . . . . . 6
54elint2 4233 . . . . 5
65orbi2i 528 . . . 4
71, 3, 63bitr4ri 286 . . 3
87abbii 2587 . 2
9 df-un 3395 . 2
10 df-iin 4272 . 2
118, 9, 103eqtr4i 2503 1
 Colors of variables: wff setvar class Syntax hints:   wo 375   wceq 1452   wcel 1904  cab 2457  wral 2756   cun 3388  cint 4226  ciin 4270 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-v 3033  df-un 3395  df-int 4227  df-iin 4272 This theorem is referenced by: (None)
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