ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  iundif2ss Structured version   Unicode version

Theorem iundif2ss 3713
Description: Indexed union of class difference. Compare to theorem "De Morgan's laws" in [Enderton] p. 31. (Contributed by Jim Kingdon, 17-Aug-2018.)
Assertion
Ref Expression
iundif2ss  U_  \  C  C_ 
\  |^|_  C
Distinct variable group:   ,
Allowed substitution hints:   ()    C()

Proof of Theorem iundif2ss
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eldif 2921 . . . . . 6  \  C  C
21rexbii 2325 . . . . 5  \  C  C
3 r19.42v 2461 . . . . 5  C  C
42, 3bitri 173 . . . 4  \  C  C
5 rexnalim 2311 . . . . . 6  C  C
6 vex 2554 . . . . . . 7 
_V
7 eliin 3653 . . . . . . 7  _V  |^|_  C  C
86, 7ax-mp 7 . . . . . 6  |^|_  C  C
95, 8sylnibr 601 . . . . 5  C  |^|_  C
109anim2i 324 . . . 4  C  |^|_  C
114, 10sylbi 114 . . 3  \  C  |^|_  C
12 eliun 3652 . . 3  U_  \  C  \  C
13 eldif 2921 . . 3  \  |^|_  C  |^|_  C
1411, 12, 133imtr4i 190 . 2  U_  \  C  \  |^|_  C
1514ssriv 2943 1  U_  \  C  C_ 
\  |^|_  C
Colors of variables: wff set class
Syntax hints:   wn 3   wa 97   wb 98   wcel 1390  wral 2300  wrex 2301   _Vcvv 2551    \ cdif 2908    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-in1 544  ax-in2 545  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-bnd 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-fal 1248  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-dif 2914  df-in 2918  df-ss 2925  df-iun 3650  df-iin 3651
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator