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Theorem sstrALT2 27398
Description: Virtual deduction proof of sstr 3108, transitivity of subclasses, Theorem 6 of [Suppes] p. 23. This theorem was automatically generated from sstrALT2VD 27397 using the command file translatewithout_overwriting.cmd . It was not minimized because the automated minimization excluding duplicates generates a minimized proof which, although not directly containing any duplicates, indirectly contains a duplicate. That is, the trace back of the minimized proof contains a duplicate. This is undesirable because some step(s) of the minimized proof use the proven theorem. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sstrALT2  |-  ( ( A  C_  B  /\  B  C_  C )  ->  A  C_  C )

Proof of Theorem sstrALT2
StepHypRef Expression
1 dfss2 3092 . 2  |-  ( A 
C_  C  <->  A. x
( x  e.  A  ->  x  e.  C ) )
2 id 21 . . . . . 6  |-  ( ( A  C_  B  /\  B  C_  C )  -> 
( A  C_  B  /\  B  C_  C ) )
3 simpr 449 . . . . . 6  |-  ( ( A  C_  B  /\  B  C_  C )  ->  B  C_  C )
42, 3syl 17 . . . . 5  |-  ( ( A  C_  B  /\  B  C_  C )  ->  B  C_  C )
5 simpl 445 . . . . . . 7  |-  ( ( A  C_  B  /\  B  C_  C )  ->  A  C_  B )
62, 5syl 17 . . . . . 6  |-  ( ( A  C_  B  /\  B  C_  C )  ->  A  C_  B )
7 idd 23 . . . . . 6  |-  ( ( A  C_  B  /\  B  C_  C )  -> 
( x  e.  A  ->  x  e.  A ) )
8 ssel2 3098 . . . . . 6  |-  ( ( A  C_  B  /\  x  e.  A )  ->  x  e.  B )
96, 7, 8ee12an 1359 . . . . 5  |-  ( ( A  C_  B  /\  B  C_  C )  -> 
( x  e.  A  ->  x  e.  B ) )
10 ssel2 3098 . . . . 5  |-  ( ( B  C_  C  /\  x  e.  B )  ->  x  e.  C )
114, 9, 10ee12an 1359 . . . 4  |-  ( ( A  C_  B  /\  B  C_  C )  -> 
( x  e.  A  ->  x  e.  C ) )
1211idi 2 . . 3  |-  ( ( A  C_  B  /\  B  C_  C )  -> 
( x  e.  A  ->  x  e.  C ) )
1312alrimiv 2012 . 2  |-  ( ( A  C_  B  /\  B  C_  C )  ->  A. x ( x  e.  A  ->  x  e.  C ) )
14 bi2 191 . 2  |-  ( ( A  C_  C  <->  A. x
( x  e.  A  ->  x  e.  C ) )  ->  ( A. x ( x  e.  A  ->  x  e.  C )  ->  A  C_  C ) )
151, 13, 14mpsyl 61 1  |-  ( ( A  C_  B  /\  B  C_  C )  ->  A  C_  C )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360   A.wal 1532    e. wcel 1621    C_ wss 3078
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-in 3085  df-ss 3089
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