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Theorem iuniin 4095
 Description: Law combining indexed union with indexed intersection. Eq. 14 in [KuratowskiMostowski] p. 109. This theorem also appears as the last example at http://en.wikipedia.org/wiki/Union%5F%28set%5Ftheory%29. (Contributed by NM, 17-Aug-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
iuniin
Distinct variable groups:   ,   ,   ,
Allowed substitution hints:   ()   ()   (,)

Proof of Theorem iuniin
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 r19.12 2811 . . . 4
2 vex 2951 . . . . . 6
3 eliin 4090 . . . . . 6
42, 3ax-mp 8 . . . . 5
54rexbii 2722 . . . 4
6 eliun 4089 . . . . 5
76ralbii 2721 . . . 4
81, 5, 73imtr4i 258 . . 3
9 eliun 4089 . . 3
10 eliin 4090 . . . 4
112, 10ax-mp 8 . . 3
128, 9, 113imtr4i 258 . 2
1312ssriv 3344 1
 Colors of variables: wff set class Syntax hints:   wb 177   wcel 1725  wral 2697  wrex 2698  cvv 2948   wss 3312  ciun 4085  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-rex 2703  df-v 2950  df-in 3319  df-ss 3326  df-iun 4087  df-iin 4088
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