Theorem psstr 3711

Description: Transitive law for proper subclass. Theorem 9 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.)

Assertion

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

Proof of Theorem psstr

Step	Hyp	Ref	Expression
1		pssss 3702	. . 3 |- ( A C. B -> A C_ B )
2		pssss 3702	. . 3 |- ( B C. C -> B C_ C )
3	1, 2	sylan9ss 3616	. 2 |- ( ( A C. B /\ B C. C ) -> A C_ C )
4		pssn2lp 3708	. . . 4 |- -. ( C C. B /\ B C. C )
5		psseq1 3694	. . . . 5 |- ( A = C -> ( A C. B <-> C C. B ) )
6	5	anbi1d 741	. . . 4 |- ( A = C -> ( ( A C. B /\ B C. C ) <-> ( C C. B /\ B C. C ) ) )
7	4, 6	mtbiri 317	. . 3 |- ( A = C -> -. ( A C. B /\ B C. C ) )
8	7	con2i 134	. 2 |- ( ( A C. B /\ B C. C ) -> -. A = C )
9		dfpss2 3692	. 2 |- ( A C. C <-> ( A C_ C /\ -. A = C ) )
10	3, 8, 9	sylanbrc 698	1 |- ( ( A C. B /\ B C. C ) -> A C. C )

Syntax hints:   -. wn 3   -> wi 4   /\ 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:  sspsstr  3712  psssstr  3713  psstrd  3714  porpss  6941  inf3lem5  8529  ltsopr  9854