Theorem sspsstr 2715

Description: Transitive law for subclass and proper subclass.

Assertion

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

Proof of Theorem sspsstr

Step	Hyp	Ref	Expression
1		psstr 2714	. . . . 5 |- ((A C. B /\ B C. C) -> A C. C)
2	1	ex 402	. . . 4 |- (A C. B -> (B C. C -> A C. C))
3		psseq1 2697	. . . . 5 |- (A = B -> (A C. C <-> B C. C))
4	3	biimprd 171	. . . 4 |- (A = B -> (B C. C -> A C. C))
5	2, 4	jaoi 368	. . 3 |- ((A C. B \/ A = B) -> (B C. C -> A C. C))
6	5	imp 377	. 2 |- (((A C. B \/ A = B) /\ B C. C) -> A C. C)
7		sspss 2707	. 2 |- (A C_ B <-> (A C. B \/ A = B))
8	6, 7	sylanb 498	1 |- ((A C_ B /\ B C. C) -> A C. C)

Syntax hints:   -> wi 3   \/ wo 239   /\ wa 240   = wceq 1298   C_ wss 2593   C. wpss 2594

This theorem is referenced by:  ordtr2 3708  php 5607  ltexprlem2 6295  suplem1pr 6313

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-10 1308  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-in 2603  df-ss 2605  df-pss 2607

Copyright terms: Public domain