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Theorem ssuni 4038
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 2498 . . . . . . 7  |-  ( x  =  B  ->  (
y  e.  x  <->  y  e.  B ) )
21imbi1d 310 . . . . . 6  |-  ( x  =  B  ->  (
( y  e.  x  ->  y  e.  U. C
)  <->  ( y  e.  B  ->  y  e.  U. C ) ) )
3 elunii 4021 . . . . . . 7  |-  ( ( y  e.  x  /\  x  e.  C )  ->  y  e.  U. C
)
43expcom 426 . . . . . 6  |-  ( x  e.  C  ->  (
y  e.  x  -> 
y  e.  U. C
) )
52, 4vtoclga 3018 . . . . 5  |-  ( B  e.  C  ->  (
y  e.  B  -> 
y  e.  U. C
) )
65imim2d 51 . . . 4  |-  ( B  e.  C  ->  (
( y  e.  A  ->  y  e.  B )  ->  ( y  e.  A  ->  y  e.  U. C ) ) )
76alimdv 1632 . . 3  |-  ( B  e.  C  ->  ( A. y ( y  e.  A  ->  y  e.  B )  ->  A. y
( y  e.  A  ->  y  e.  U. C
) ) )
8 dfss2 3338 . . 3  |-  ( A 
C_  B  <->  A. y
( y  e.  A  ->  y  e.  B ) )
9 dfss2 3338 . . 3  |-  ( A 
C_  U. C  <->  A. y
( y  e.  A  ->  y  e.  U. C
) )
107, 8, 93imtr4g 263 . 2  |-  ( B  e.  C  ->  ( A  C_  B  ->  A  C_ 
U. C ) )
1110impcom 421 1  |-  ( ( A  C_  B  /\  B  e.  C )  ->  A  C_  U. C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360   A.wal 1550    = wceq 1653    e. wcel 1726    C_ wss 3321   U.cuni 4016
This theorem is referenced by:  elssuni  4044  uniss2  4047  ssorduni  4767  filssufilg  17944  alexsubALTlem2  18080  utoptop  18265
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-v 2959  df-in 3328  df-ss 3335  df-uni 4017
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