Theorem psssstr 3550

Description: Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.)

Assertion

Ref	Expression
psssstr	|- ( ( A C. B /\ B C_ C ) -> A C. C )

Proof of Theorem psssstr

Step	Hyp	Ref	Expression
1		sspss 3543	. 2 |- ( B C_ C <-> ( B C. C \/ B = C ) )
2		psstr 3548	. . . . 5 |- ( ( A C. B /\ B C. C ) -> A C. C )
3	2	ex 440	. . . 4 |- ( A C. B -> ( B C. C -> A C. C ) )
4		psseq2 3532	. . . . 5 |- ( B = C -> ( A C. B <-> A C. C ) )
5	4	biimpcd 232	. . . 4 |- ( A C. B -> ( B = C -> A C. C ) )
6	3, 5	jaod 386	. . 3 |- ( A C. B -> ( ( B C. C \/ B = C ) -> A C. C ) )
7	6	imp 435	. 2 |- ( ( A C. B /\ ( B C. C \/ B = C ) ) -> A C. C )
8	1, 7	sylan2b 482	1 |- ( ( A C. B /\ B C_ C ) -> A C. C )

Syntax hints:   -> wi 4   \/ wo 374   /\ wa 375   = wceq 1454   C_ wss 3415   C. wpss 3416

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-ne 2634  df-in 3422  df-ss 3429  df-pss 3431

This theorem is referenced by:  psssstrd  3553  suplem1pr  9502  atexch  28082  bj-2upln0  31661  bj-2upln1upl  31662