Theorem sspsstr 3375

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

Assertion

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

Proof of Theorem sspsstr

Step	Hyp	Ref	Expression
1		sspss 3369	. 2 |- (A C_ B <-> (A C. B \/ A = B))
2		psstr 3374	. . . . 5 |- ((A C. B /\ B C. C) -> A C. C)
3	2	ex 423	. . . 4 |- (A C. B -> (B C. C -> A C. C))
4		psseq1 3357	. . . . 5 |- (A = B -> (A C. C <-> B C. C))
5	4	biimprd 214	. . . 4 |- (A = B -> (B C. C -> A C. C))
6	3, 5	jaoi 368	. . 3 |- ((A C. B \/ A = B) -> (B C. C -> A C. C))
7	6	imp 418	. 2 |- (((A C. B \/ A = B) /\ B C. C) -> A C. C)
8	1, 7	sylanb 458	1 |- ((A C_ B /\ B C. C) -> A C. C)

Syntax hints:   -> wi 4   \/ wo 357   /\ wa 358   = wceq 1642   C_ wss 3258   C. wpss 3259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-pss 3262

This theorem is referenced by:  sspsstrd  3378