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Theorem sstr2 3108
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 3095 . . . 4  |-  ( A 
C_  B  ->  (
x  e.  A  ->  x  e.  B )
)
21imim1d 75 . . 3  |-  ( A 
C_  B  ->  (
( x  e.  B  ->  x  e.  C )  ->  ( x  e.  A  ->  x  e.  C ) ) )
32alimdv 1852 . 2  |-  ( A 
C_  B  ->  ( A. x ( x  e.  B  ->  x  e.  C )  ->  A. x
( x  e.  A  ->  x  e.  C ) ) )
4 dfss2 3090 . 2  |-  ( B 
C_  C  <->  A. x
( x  e.  B  ->  x  e.  C ) )
5 dfss2 3090 . 2  |-  ( A 
C_  C  <->  A. x
( x  e.  A  ->  x  e.  C ) )
63, 4, 53imtr4g 204 1  |-  ( A 
C_  B  ->  ( B  C_  C  ->  A  C_  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1330    e. wcel 1481    C_ wss 3075
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-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-in 3081  df-ss 3088
This theorem is referenced by:  sstr  3109  sstri  3110  sseq1  3124  sseq2  3125  ssun3  3245  ssun4  3246  ssinss1  3309  ssdisj  3423  triun  4046  trintssm  4049  sspwb  4145  exss  4156  relss  4633  funss  5149  funimass2  5208  fss  5291  fiintim  6824  sbthlem2  6853  sbthlemi3  6854  sbthlemi6  6857  tgss  12269  tgcl  12270  tgss3  12284  clsss  12324  neiss  12356  ssnei2  12363  cnpnei  12425  cnptopco  12428  cnptoprest  12445  txcnp  12477  neibl  12697  metcnp3  12717  bj-nntrans  13318
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