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Theorem iuniin 4253
 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 2893 . . . 4
2 vex 3025 . . . . . 6
3 eliin 4248 . . . . . 6
42, 3ax-mp 5 . . . . 5
54rexbii 2866 . . . 4
6 eliun 4247 . . . . 5
76ralbii 2796 . . . 4
81, 5, 73imtr4i 269 . . 3
9 eliun 4247 . . 3
10 eliin 4248 . . . 4
112, 10ax-mp 5 . . 3
128, 9, 113imtr4i 269 . 2
1312ssriv 3411 1
 Colors of variables: wff setvar class Syntax hints:   wb 187   wcel 1872  wral 2714  wrex 2715  cvv 3022   wss 3379  ciun 4242  ciin 4243 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408 This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ral 2719  df-rex 2720  df-v 3024  df-in 3386  df-ss 3393  df-iun 4244  df-iin 4245 This theorem is referenced by: (None)
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