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Theorem ss2iun 4331
Description: Subclass theorem for indexed union. (Contributed by NM, 26-Nov-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
ss2iun  |-  ( A. x  e.  A  B  C_  C  ->  U_ x  e.  A  B  C_  U_ x  e.  A  C )

Proof of Theorem ss2iun
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ssel 3483 . . . . 5  |-  ( B 
C_  C  ->  (
y  e.  B  -> 
y  e.  C ) )
21ralimi 2836 . . . 4  |-  ( A. x  e.  A  B  C_  C  ->  A. x  e.  A  ( y  e.  B  ->  y  e.  C ) )
3 rexim 2908 . . . 4  |-  ( A. x  e.  A  (
y  e.  B  -> 
y  e.  C )  ->  ( E. x  e.  A  y  e.  B  ->  E. x  e.  A  y  e.  C )
)
42, 3syl 16 . . 3  |-  ( A. x  e.  A  B  C_  C  ->  ( E. x  e.  A  y  e.  B  ->  E. x  e.  A  y  e.  C ) )
5 eliun 4320 . . 3  |-  ( y  e.  U_ x  e.  A  B  <->  E. x  e.  A  y  e.  B )
6 eliun 4320 . . 3  |-  ( y  e.  U_ x  e.  A  C  <->  E. x  e.  A  y  e.  C )
74, 5, 63imtr4g 270 . 2  |-  ( A. x  e.  A  B  C_  C  ->  ( y  e.  U_ x  e.  A  B  ->  y  e.  U_ x  e.  A  C
) )
87ssrdv 3495 1  |-  ( A. x  e.  A  B  C_  C  ->  U_ x  e.  A  B  C_  U_ x  e.  A  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1804   A.wral 2793   E.wrex 2794    C_ wss 3461   U_ciun 4315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ral 2798  df-rex 2799  df-v 3097  df-in 3468  df-ss 3475  df-iun 4317
This theorem is referenced by:  iuneq2  4332  oawordri  7201  omwordri  7223  oewordri  7243  oeworde  7244  r1val1  8207  cfslb2n  8651  imasaddvallem  14908  dprdss  17055  tgcmp  19879  txcmplem1  20120  txcmplem2  20121  xkococnlem  20138  alexsubALT  20529  ptcmplem3  20532  metnrmlem2  21342  uniiccvol  21967  dvfval  22279  filnetlem3  30174  sstotbnd2  30246  equivtotbnd  30250  bnj1145  33917  bnj1136  33921
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