Theorem psssstr 3563

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 3556	. 2 |- ( B C_ C <-> ( B C. C \/ B = C ) )
2		psstr 3561	. . . . 5 |- ( ( A C. B /\ B C. C ) -> A C. C )
3	2	ex 434	. . . 4 |- ( A C. B -> ( B C. C -> A C. C ) )
4		psseq2 3545	. . . . 5 |- ( B = C -> ( A C. B <-> A C. C ) )
5	4	biimpcd 224	. . . 4 |- ( A C. B -> ( B = C -> A C. C ) )
6	3, 5	jaod 380	. . 3 |- ( A C. B -> ( ( B C. C \/ B = C ) -> A C. C ) )
7	6	imp 429	. 2 |- ( ( A C. B /\ ( B C. C \/ B = C ) ) -> A C. C )
8	1, 7	sylan2b 475	1 |- ( ( A C. B /\ B C_ C ) -> A C. C )

Syntax hints:   -> wi 4   \/ wo 368   /\ wa 369   = wceq 1370   C_ wss 3429   C. wpss 3430

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 1952  ax-ext 2430

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-ne 2646  df-in 3436  df-ss 3443  df-pss 3445

This theorem is referenced by:  psssstrd  3566  suplem1pr  9325  atexch  25930  bj-2upln0  32819  bj-2upln1upl  32820