Theorem sspsstr 3050

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 3039	. . 3 |- ( B C. C -> B C_ C )
3	1, 2	sylan9ss 2958	. 2 |- ( ( A C_ B /\ B C. C ) -> A C_ C )
4		sspssn 3048	. . . 4 |- -. ( C C_ B /\ B C. C )
5		sseq1 2966	. . . . 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 600	. . 3 |- ( A = C -> -. ( A C_ B /\ B C. C ) )
8	7	con2i 557	. 2 |- ( ( A C_ B /\ B C. C ) -> -. A = C )
9		dfpss2 3029	. 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 1243   C_ wss 2917   C. wpss 2918

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 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-ne 2206  df-in 2924  df-ss 2931  df-pss 2933

This theorem is referenced by:  sspsstrd  3053