Theorem sspsstr 3044

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		id 19	. . 3 |- (A C_ B -> A C_ B)
2		pssss 3033	. . 3 |- (B C. C -> B C_ C)
3	1, 2	sylan9ss 2952	. 2 |- ((A C_ B /\ B C. C) -> A C_ C)
4		sspssn 3042	. . . 4 |- -. ( C C_ B /\ B C. C)
5		sseq1 2960	. . . . 5 |- (A = C -> (A C_ B <-> C C_ B))
6	5	anbi1d 438	. . . 4 |- (A = C -> ((A C_ B /\ B C. C) <-> ( C C_ B /\ B C. C)))
7	4, 6	mtbiri 599	. . 3 |- (A = C -> -. (A C_ B /\ B C. C))
8	7	con2i 557	. 2 |- ((A C_ B /\ B C. C) -> -. A = C)
9		dfpss2 3023	. 2 |- (A C. C <-> (A C_ C /\ -. A = C))
10	3, 8, 9	sylanbrc 394	1 |- ((A C_ B /\ B C. C) -> A C. C)

Syntax hints:  -. wn 3   -> wi 4   /\ wa 97   = wceq 1242   C_ wss 2911   C. wpss 2912

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-ne 2203  df-in 2918  df-ss 2925  df-pss 2927

This theorem is referenced by:  sspsstrd  3047