Theorem psssstr 3596

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 3589	. 2 |- ( B C_ C <-> ( B C. C \/ B = C ) )
2		psstr 3594	. . . . 5 |- ( ( A C. B /\ B C. C ) -> A C. C )
3	2	ex 432	. . . 4 |- ( A C. B -> ( B C. C -> A C. C ) )
4		psseq2 3578	. . . . 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 378	. . 3 |- ( A C. B -> ( ( B C. C \/ B = C ) -> A C. C ) )
7	6	imp 427	. 2 |- ( ( A C. B /\ ( B C. C \/ B = C ) ) -> A C. C )
8	1, 7	sylan2b 473	1 |- ( ( A C. B /\ B C_ C ) -> A C. C )

Syntax hints:   -> wi 4   \/ wo 366   /\ wa 367   = wceq 1398   C_ wss 3461   C. wpss 3462

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-ne 2651  df-in 3468  df-ss 3475  df-pss 3477

This theorem is referenced by:  psssstrd  3599  suplem1pr  9419  atexch  27498  bj-2upln0  34982  bj-2upln1upl  34983