ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  iuniin Structured version   Unicode version

Theorem iuniin 3658
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  U_  |^|_  C  C_  |^|_  U_  C
Distinct variable groups:   ,   ,   ,
Allowed substitution hints:   ()   ()    C(,)

Proof of Theorem iuniin
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 r19.12 2416 . . . 4  C  C
2 vex 2554 . . . . . 6 
_V
3 eliin 3653 . . . . . 6  _V  |^|_  C  C
42, 3ax-mp 7 . . . . 5  |^|_  C  C
54rexbii 2325 . . . 4  |^|_  C  C
6 eliun 3652 . . . . 5  U_  C  C
76ralbii 2324 . . . 4  U_  C  C
81, 5, 73imtr4i 190 . . 3  |^|_  C  U_  C
9 eliun 3652 . . 3  U_  |^|_  C  |^|_  C
10 eliin 3653 . . . 4  _V  |^|_  U_  C  U_  C
112, 10ax-mp 7 . . 3  |^|_  U_  C  U_  C
128, 9, 113imtr4i 190 . 2  U_  |^|_  C  |^|_  U_  C
1312ssriv 2943 1  U_  |^|_  C  C_  |^|_  U_  C
Colors of variables: wff set class
Syntax hints:   wb 98   wcel 1390  wral 2300  wrex 2301   _Vcvv 2551    C_ wss 2911   U_ciun 3648   |^|_ciin 3649
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-in 2918  df-ss 2925  df-iun 3650  df-iin 3651
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator