Theorem sstrALT2 31884

Description: Virtual deduction proof of sstr 3467, transitivity of subclasses, Theorem 6 of [Suppes] p. 23. This theorem was automatically generated from sstrALT2VD 31883 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 3448	. 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 3454	. . . . . 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 3454	. . . . 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 31466	. . 3 |- ( ( A C_ B /\ B C_ C ) -> ( x e. A -> x e. C ) )
13	12	alrimiv 1686	. 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 1368   e. wcel 1758   C_ wss 3431

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2438  df-cleq 2444  df-clel 2447  df-in 3438  df-ss 3445

This theorem is referenced by: (None)