Theorem sstr 2953

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 2952	. 2 |- ( A C_ B -> ( B C_ C -> A C_ C ) )
2	1	imp 115	1 |- ( ( A C_ B /\ B C_ C ) -> A C_ C )

Syntax hints:   -> wi 4   /\ wa 97   C_ wss 2917

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-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-in 2924  df-ss 2931

This theorem is referenced by:  sstrd  2955  sylan9ss  2958  ssdifss  3074  uneqin  3188  ssindif0im  3281  undifss  3303  ssrnres  4763  relrelss  4844  fco  5056  fssres  5066  ssimaex  5234  tpostpos2  5880  smores  5907  iccsupr  8835