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Theorem sstrALT2 28611
Description: Virtual deduction proof of sstr 3187, transitivity of subclasses, Theorem 6 of [Suppes] p. 23. This theorem was automatically generated from sstrALT2VD 28610 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
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 dfss2 3169 . 2  |-  ( A 
C_  C  <->  A. x
( x  e.  A  ->  x  e.  C ) )
2 id 19 . . . . . 6  |-  ( ( A  C_  B  /\  B  C_  C )  -> 
( A  C_  B  /\  B  C_  C ) )
3 simpr 447 . . . . . 6  |-  ( ( A  C_  B  /\  B  C_  C )  ->  B  C_  C )
42, 3syl 15 . . . . 5  |-  ( ( A  C_  B  /\  B  C_  C )  ->  B  C_  C )
5 simpl 443 . . . . . . 7  |-  ( ( A  C_  B  /\  B  C_  C )  ->  A  C_  B )
62, 5syl 15 . . . . . 6  |-  ( ( A  C_  B  /\  B  C_  C )  ->  A  C_  B )
7 idd 21 . . . . . 6  |-  ( ( A  C_  B  /\  B  C_  C )  -> 
( x  e.  A  ->  x  e.  A ) )
8 ssel2 3175 . . . . . 6  |-  ( ( A  C_  B  /\  x  e.  A )  ->  x  e.  B )
96, 7, 8ee12an 1353 . . . . 5  |-  ( ( A  C_  B  /\  B  C_  C )  -> 
( x  e.  A  ->  x  e.  B ) )
10 ssel2 3175 . . . . 5  |-  ( ( B  C_  C  /\  x  e.  B )  ->  x  e.  C )
114, 9, 10ee12an 1353 . . . 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 1617 . 2  |-  ( ( A  C_  B  /\  B  C_  C )  ->  A. x ( x  e.  A  ->  x  e.  C ) )
14 bi2 189 . 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 59 1  |-  ( ( A  C_  B  /\  B  C_  C )  ->  A  C_  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527    e. wcel 1684    C_ wss 3152
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-in 3159  df-ss 3166
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