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Theorem ssuni 4224
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 2527 . . . . . . 7  |-  ( x  =  B  ->  (
y  e.  x  <->  y  e.  B ) )
21imbi1d 317 . . . . . 6  |-  ( x  =  B  ->  (
( y  e.  x  ->  y  e.  U. C
)  <->  ( y  e.  B  ->  y  e.  U. C ) ) )
3 elunii 4207 . . . . . . 7  |-  ( ( y  e.  x  /\  x  e.  C )  ->  y  e.  U. C
)
43expcom 435 . . . . . 6  |-  ( x  e.  C  ->  (
y  e.  x  -> 
y  e.  U. C
) )
52, 4vtoclga 3142 . . . . 5  |-  ( B  e.  C  ->  (
y  e.  B  -> 
y  e.  U. C
) )
65imim2d 52 . . . 4  |-  ( B  e.  C  ->  (
( y  e.  A  ->  y  e.  B )  ->  ( y  e.  A  ->  y  e.  U. C ) ) )
76alimdv 1676 . . 3  |-  ( B  e.  C  ->  ( A. y ( y  e.  A  ->  y  e.  B )  ->  A. y
( y  e.  A  ->  y  e.  U. C
) ) )
8 dfss2 3456 . . 3  |-  ( A 
C_  B  <->  A. y
( y  e.  A  ->  y  e.  B ) )
9 dfss2 3456 . . 3  |-  ( A 
C_  U. C  <->  A. y
( y  e.  A  ->  y  e.  U. C
) )
107, 8, 93imtr4g 270 . 2  |-  ( B  e.  C  ->  ( A  C_  B  ->  A  C_ 
U. C ) )
1110impcom 430 1  |-  ( ( A  C_  B  /\  B  e.  C )  ->  A  C_  U. C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   A.wal 1368    = wceq 1370    e. wcel 1758    C_ wss 3439   U.cuni 4202
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3080  df-in 3446  df-ss 3453  df-uni 4203
This theorem is referenced by:  elssuni  4232  uniss2  4235  ssorduni  6510  filssufilg  19619  alexsubALTlem2  19755  utoptop  19944
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