Theorem sstrALT2 32715

Description: Virtual deduction proof of sstr 3512, transitivity of subclasses, Theorem 6 of [Suppes] p. 23. This theorem was automatically generated from sstrALT2VD 32714 using the command file translate_without_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.

Step	Hyp	Ref	Expression
1		dfss2 3493	. 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 461	. . . . . 6 |- ( ( A C_ B /\ B C_ C ) -> B C_ C )
4	2, 3	syl 16	. . . . 5 |- ( ( A C_ B /\ B C_ C ) -> B C_ C )
5		simpl 457	. . . . . . 7 |- ( ( A C_ B /\ B C_ C ) -> A C_ B )
6	2, 5	syl 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 3499	. . . . . 6 |- ( ( A C_ B /\ x e. A ) -> x e. B )
9	6, 7, 8	syl6an 545	. . . . 5 |- ( ( A C_ B /\ B C_ C ) -> ( x e. A -> x e. B ) )
10		ssel2 3499	. . . . 5 |- ( ( B C_ C /\ x e. B ) -> x e. C )
11	4, 9, 10	syl6an 545	. . . 4 |- ( ( A C_ B /\ B C_ C ) -> ( x e. A -> x e. C ) )
12	11	idiALT 32297	. . 3 |- ( ( A C_ B /\ B C_ C ) -> ( x e. A -> x e. C ) )
13	12	alrimiv 1695	. 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 ) )
15	1, 13, 14	mpsyl 63	1 |- ( ( A C_ B /\ B C_ C ) -> A C_ C )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   A.wal 1377   e. wcel 1767   C_ wss 3476

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-in 3483  df-ss 3490

This theorem is referenced by: (None)