ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  iunss2 Unicode version

Theorem iunss2 3702
Description: A subclass condition on the members of two indexed classes 
C ( x ) and  D ( y ) that implies a subclass relation on their indexed unions. Generalization of Proposition 8.6 of [TakeutiZaring] p. 59. Compare uniss2 3611. (Contributed by NM, 9-Dec-2004.)
Assertion
Ref Expression
iunss2  |-  ( A. x  e.  A  E. y  e.  B  C  C_  D  ->  U_ x  e.  A  C  C_  U_ y  e.  B  D )
Distinct variable groups:    x, y    x, B    y, C    x, D
Allowed substitution hints:    A( x, y)    B( y)    C( x)    D( y)

Proof of Theorem iunss2
StepHypRef Expression
1 ssiun 3699 . . 3  |-  ( E. y  e.  B  C  C_  D  ->  C  C_  U_ y  e.  B  D )
21ralimi 2384 . 2  |-  ( A. x  e.  A  E. y  e.  B  C  C_  D  ->  A. x  e.  A  C  C_  U_ y  e.  B  D )
3 iunss 3698 . 2  |-  ( U_ x  e.  A  C  C_ 
U_ y  e.  B  D 
<-> 
A. x  e.  A  C  C_  U_ y  e.  B  D )
42, 3sylibr 137 1  |-  ( A. x  e.  A  E. y  e.  B  C  C_  D  ->  U_ x  e.  A  C  C_  U_ y  e.  B  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wral 2306   E.wrex 2307    C_ wss 2917   U_ciun 3657
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-in 2924  df-ss 2931  df-iun 3659
This theorem is referenced by:  iunxdif2  3705  rdgss  5970
  Copyright terms: Public domain W3C validator