Theorem sstr 3033

Description: Transitivity of subclasses. Theorem 6 of [Suppes] p. 23. (Contributed by NM, 5-Sep-2003.)

Assertion

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

Proof of Theorem sstr

Step	Hyp	Ref	Expression
1		sstr2 3032	. 2 |- ( A C_ B -> ( B C_ C -> A C_ C ) )
2	1	imp 122	1 |- ( ( A C_ B /\ B C_ C ) -> A C_ C )

Syntax hints:   -> wi 4   /\ wa 102   C_ wss 2999

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-in 3005  df-ss 3012

This theorem is referenced by:  sstrd  3035  sylan9ss  3038  ssdifss  3130  uneqin  3250  ssindif0im  3342  undifss  3363  ssrnres  4873  relrelss  4957  fco  5176  fssres  5186  ssimaex  5365  tpostpos2  6030  smores  6057  pmss12g  6430  fidcenumlemr  6662  iccsupr  9382  fimaxq  10231  fsum2d  10825  fsumabs  10855