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Theorem sstrALT2 34035
Description: Virtual deduction proof of sstr 3497, transitivity of subclasses, Theorem 6 of [Suppes] p. 23. This theorem was automatically generated from sstrALT2VD 34034 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 3478 . 2  |-  ( A 
C_  C  <->  A. x
( x  e.  A  ->  x  e.  C ) )
2 id 22 . . . . . 6  |-  ( ( A  C_  B  /\  B  C_  C )  -> 
( A  C_  B  /\  B  C_  C ) )
3 simpr 459 . . . . . 6  |-  ( ( A  C_  B  /\  B  C_  C )  ->  B  C_  C )
42, 3syl 16 . . . . 5  |-  ( ( A  C_  B  /\  B  C_  C )  ->  B  C_  C )
5 simpl 455 . . . . . . 7  |-  ( ( A  C_  B  /\  B  C_  C )  ->  A  C_  B )
62, 5syl 16 . . . . . 6  |-  ( ( A  C_  B  /\  B  C_  C )  ->  A  C_  B )
7 idd 24 . . . . . 6  |-  ( ( A  C_  B  /\  B  C_  C )  -> 
( x  e.  A  ->  x  e.  A ) )
8 ssel2 3484 . . . . . 6  |-  ( ( A  C_  B  /\  x  e.  A )  ->  x  e.  B )
96, 7, 8syl6an 543 . . . . 5  |-  ( ( A  C_  B  /\  B  C_  C )  -> 
( x  e.  A  ->  x  e.  B ) )
10 ssel2 3484 . . . . 5  |-  ( ( B  C_  C  /\  x  e.  B )  ->  x  e.  C )
114, 9, 10syl6an 543 . . . 4  |-  ( ( A  C_  B  /\  B  C_  C )  -> 
( x  e.  A  ->  x  e.  C ) )
1211idiALT 33605 . . 3  |-  ( ( A  C_  B  /\  B  C_  C )  -> 
( x  e.  A  ->  x  e.  C ) )
1312alrimiv 1724 . 2  |-  ( ( A  C_  B  /\  B  C_  C )  ->  A. x ( x  e.  A  ->  x  e.  C ) )
14 bi2 198 . 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 63 1  |-  ( ( A  C_  B  /\  B  C_  C )  ->  A  C_  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367   A.wal 1396    e. wcel 1823    C_ wss 3461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-in 3468  df-ss 3475
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator