MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ssiun Structured version   Visualization version   Unicode version

Theorem ssiun 4334
Description: Subset implication for an indexed union. (Contributed by NM, 3-Sep-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
ssiun  |-  ( E. x  e.  A  C  C_  B  ->  C  C_  U_ x  e.  A  B )
Distinct variable group:    x, C
Allowed substitution hints:    A( x)    B( x)

Proof of Theorem ssiun
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ssel 3438 . . . . 5  |-  ( C 
C_  B  ->  (
y  e.  C  -> 
y  e.  B ) )
21reximi 2867 . . . 4  |-  ( E. x  e.  A  C  C_  B  ->  E. x  e.  A  ( y  e.  C  ->  y  e.  B ) )
3 r19.37v 2952 . . . 4  |-  ( E. x  e.  A  ( y  e.  C  -> 
y  e.  B )  ->  ( y  e.  C  ->  E. x  e.  A  y  e.  B ) )
42, 3syl 17 . . 3  |-  ( E. x  e.  A  C  C_  B  ->  ( y  e.  C  ->  E. x  e.  A  y  e.  B ) )
5 eliun 4297 . . 3  |-  ( y  e.  U_ x  e.  A  B  <->  E. x  e.  A  y  e.  B )
64, 5syl6ibr 235 . 2  |-  ( E. x  e.  A  C  C_  B  ->  ( y  e.  C  ->  y  e. 
U_ x  e.  A  B ) )
76ssrdv 3450 1  |-  ( E. x  e.  A  C  C_  B  ->  C  C_  U_ x  e.  A  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1898   E.wrex 2750    C_ wss 3416   U_ciun 4292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442
This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ral 2754  df-rex 2755  df-v 3059  df-in 3423  df-ss 3430  df-iun 4294
This theorem is referenced by:  iunss2  4337  iunpwss  4385  iunpw  6632  wfrdmcl  7070  onfununi  7086  oen0  7313  trcl  8238  rtrclreclem1  13170  rtrclreclem2  13171  trpredtr  30520  dftrpred3g  30523  frrlem5e  30571
  Copyright terms: Public domain W3C validator