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Theorem ssuni 3806
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 2228 . . . . . . 7  |-  ( x  =  B  ->  (
y  e.  x  <->  y  e.  B ) )
21imbi1d 230 . . . . . 6  |-  ( x  =  B  ->  (
( y  e.  x  ->  y  e.  U. C
)  <->  ( y  e.  B  ->  y  e.  U. C ) ) )
3 elunii 3789 . . . . . . 7  |-  ( ( y  e.  x  /\  x  e.  C )  ->  y  e.  U. C
)
43expcom 115 . . . . . 6  |-  ( x  e.  C  ->  (
y  e.  x  -> 
y  e.  U. C
) )
52, 4vtoclga 2788 . . . . 5  |-  ( B  e.  C  ->  (
y  e.  B  -> 
y  e.  U. C
) )
65imim2d 54 . . . 4  |-  ( B  e.  C  ->  (
( y  e.  A  ->  y  e.  B )  ->  ( y  e.  A  ->  y  e.  U. C ) ) )
76alimdv 1866 . . 3  |-  ( B  e.  C  ->  ( A. y ( y  e.  A  ->  y  e.  B )  ->  A. y
( y  e.  A  ->  y  e.  U. C
) ) )
8 dfss2 3127 . . 3  |-  ( A 
C_  B  <->  A. y
( y  e.  A  ->  y  e.  B ) )
9 dfss2 3127 . . 3  |-  ( A 
C_  U. C  <->  A. y
( y  e.  A  ->  y  e.  U. C
) )
107, 8, 93imtr4g 204 . 2  |-  ( B  e.  C  ->  ( A  C_  B  ->  A  C_ 
U. C ) )
1110impcom 124 1  |-  ( ( A  C_  B  /\  B  e.  C )  ->  A  C_  U. C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   A.wal 1340    = wceq 1342    e. wcel 2135    C_ wss 3112   U.cuni 3784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146
This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2724  df-in 3118  df-ss 3125  df-uni 3785
This theorem is referenced by:  elssuni  3812  uniss2  3815  ssorduni  4459
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