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Theorem ssuni 3849
Description: Subclass relationship for class union. (Contributed by NM, 24-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
ssuni  |-  ( ( A  C_  B  /\  B  e.  C )  ->  A  C_  U. C )

Proof of Theorem ssuni
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq2 2344 . . . . . . 7  |-  ( x  =  B  ->  (
y  e.  x  <->  y  e.  B ) )
21imbi1d 308 . . . . . 6  |-  ( x  =  B  ->  (
( y  e.  x  ->  y  e.  U. C
)  <->  ( y  e.  B  ->  y  e.  U. C ) ) )
3 elunii 3832 . . . . . . 7  |-  ( ( y  e.  x  /\  x  e.  C )  ->  y  e.  U. C
)
43expcom 424 . . . . . 6  |-  ( x  e.  C  ->  (
y  e.  x  -> 
y  e.  U. C
) )
52, 4vtoclga 2849 . . . . 5  |-  ( B  e.  C  ->  (
y  e.  B  -> 
y  e.  U. C
) )
65imim2d 48 . . . 4  |-  ( B  e.  C  ->  (
( y  e.  A  ->  y  e.  B )  ->  ( y  e.  A  ->  y  e.  U. C ) ) )
76alimdv 1607 . . 3  |-  ( B  e.  C  ->  ( A. y ( y  e.  A  ->  y  e.  B )  ->  A. y
( y  e.  A  ->  y  e.  U. C
) ) )
8 dfss2 3169 . . 3  |-  ( A 
C_  B  <->  A. y
( y  e.  A  ->  y  e.  B ) )
9 dfss2 3169 . . 3  |-  ( A 
C_  U. C  <->  A. y
( y  e.  A  ->  y  e.  U. C
) )
107, 8, 93imtr4g 261 . 2  |-  ( B  e.  C  ->  ( A  C_  B  ->  A  C_ 
U. C ) )
1110impcom 419 1  |-  ( ( A  C_  B  /\  B  e.  C )  ->  A  C_  U. C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1527    = wceq 1623    e. wcel 1684    C_ wss 3152   U.cuni 3827
This theorem is referenced by:  elssuni  3855  uniss2  3858  ssorduni  4577  filssufilg  17606  alexsubALTlem2  17742  difelsiga  23494  toplat  25290
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-in 3159  df-ss 3166  df-uni 3828
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