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Theorem sstr2 3021
Description: Transitivity of subclasses. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
Assertion
Ref Expression
sstr2  |-  ( A 
C_  B  ->  ( B  C_  C  ->  A  C_  C ) )

Proof of Theorem sstr2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ssel 3008 . . . 4  |-  ( A 
C_  B  ->  (
x  e.  A  ->  x  e.  B )
)
21imim1d 74 . . 3  |-  ( A 
C_  B  ->  (
( x  e.  B  ->  x  e.  C )  ->  ( x  e.  A  ->  x  e.  C ) ) )
32alimdv 1804 . 2  |-  ( A 
C_  B  ->  ( A. x ( x  e.  B  ->  x  e.  C )  ->  A. x
( x  e.  A  ->  x  e.  C ) ) )
4 dfss2 3003 . 2  |-  ( B 
C_  C  <->  A. x
( x  e.  B  ->  x  e.  C ) )
5 dfss2 3003 . 2  |-  ( A 
C_  C  <->  A. x
( x  e.  A  ->  x  e.  C ) )
63, 4, 53imtr4g 203 1  |-  ( A 
C_  B  ->  ( B  C_  C  ->  A  C_  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1285    e. wcel 1436    C_ wss 2988
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-11 1440  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-in 2994  df-ss 3001
This theorem is referenced by:  sstr  3022  sstri  3023  sseq1  3036  sseq2  3037  ssun3  3154  ssun4  3155  ssinss1  3217  ssdisj  3327  triun  3926  trintssm  3929  sspwb  4019  exss  4030  relss  4495  funss  5001  funimass2  5059  fss  5138  sbthlem2  6614  sbthlemi3  6615  sbthlemi6  6618  bj-nntrans  11334
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