Theorem sspsstr 3712

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 3706	. 2 |- ( A C_ B <-> ( A C. B \/ A = B ) )
2		psstr 3711	. . . . 5 |- ( ( A C. B /\ B C. C ) -> A C. C )
3	2	ex 450	. . . 4 |- ( A C. B -> ( B C. C -> A C. C ) )
4		psseq1 3694	. . . . 5 |- ( A = B -> ( A C. C <-> B C. C ) )
5	4	biimprd 238	. . . 4 |- ( A = B -> ( B C. C -> A C. C ) )
6	3, 5	jaoi 394	. . 3 |- ( ( A C. B \/ A = B ) -> ( B C. C -> A C. C ) )
7	6	imp 445	. 2 |- ( ( ( A C. B \/ A = B ) /\ B C. C ) -> A C. C )
8	1, 7	sylanb 489	1 |- ( ( A C_ B /\ B C. C ) -> A C. C )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   C_ wss 3574   C. wpss 3575

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-ne 2795  df-in 3581  df-ss 3588  df-pss 3590

This theorem is referenced by:  sspsstrd  3715  ordtr2  5768  php  8144  canthp1lem2  9475  suplem1pr  9874  fbfinnfr  21645  ppiltx  24903