Theorem sstrALT2

Description: Virtual deduction proof of sstr , transitivity of subclasses, Theorem 6 of Suppes p. 23. This theorem was automatically generated from sstrALT2VD 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.)

	Ref	Expression
Assertion	sstrALT2	|- ( ( A C_ B /\ B C_ C ) -> A C_ C )

Proof

Step	Hyp	Ref	Expression
1		df-ss	|- ( A C_ C <-> A. x ( x e. A -> x e. C ) )
2		id	|- ( ( A C_ B /\ B C_ C ) -> ( A C_ B /\ B C_ C ) )
3		simpr	|- ( ( A C_ B /\ B C_ C ) -> B C_ C )
4	2 3	syl	|- ( ( A C_ B /\ B C_ C ) -> B C_ C )
5		simpl	|- ( ( A C_ B /\ B C_ C ) -> A C_ B )
6	2 5	syl	|- ( ( A C_ B /\ B C_ C ) -> A C_ B )
7		idd	|- ( ( A C_ B /\ B C_ C ) -> ( x e. A -> x e. A ) )
8		ssel2	|- ( ( A C_ B /\ x e. A ) -> x e. B )
9	6 7 8	syl6an	|- ( ( A C_ B /\ B C_ C ) -> ( x e. A -> x e. B ) )
10		ssel2	|- ( ( B C_ C /\ x e. B ) -> x e. C )
11	4 9 10	syl6an	|- ( ( A C_ B /\ B C_ C ) -> ( x e. A -> x e. C ) )
12	11	idiALT	|- ( ( A C_ B /\ B C_ C ) -> ( x e. A -> x e. C ) )
13	12	alrimiv	|- ( ( A C_ B /\ B C_ C ) -> A. x ( x e. A -> x e. C ) )
14		biimpr	|- ( ( 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	|- ( ( A C_ B /\ B C_ C ) -> A C_ C )