MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sspsstr Unicode version

Theorem sspsstr 3294
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
StepHypRef Expression
1 sspss 3288 . 2  |-  ( A 
C_  B  <->  ( A  C.  B  \/  A  =  B ) )
2 psstr 3293 . . . . 5  |-  ( ( A  C.  B  /\  B  C.  C )  ->  A  C.  C )
32ex 423 . . . 4  |-  ( A 
C.  B  ->  ( B  C.  C  ->  A  C.  C ) )
4 psseq1 3276 . . . . 5  |-  ( A  =  B  ->  ( A  C.  C  <->  B  C.  C ) )
54biimprd 214 . . . 4  |-  ( A  =  B  ->  ( B  C.  C  ->  A  C.  C ) )
63, 5jaoi 368 . . 3  |-  ( ( A  C.  B  \/  A  =  B )  ->  ( B  C.  C  ->  A  C.  C ) )
76imp 418 . 2  |-  ( ( ( A  C.  B  \/  A  =  B
)  /\  B  C.  C )  ->  A  C.  C )
81, 7sylanb 458 1  |-  ( ( A  C_  B  /\  B  C.  C )  ->  A  C.  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1632    C_ wss 3165    C. wpss 3166
This theorem is referenced by:  sspsstrd  3297  ordtr2  4452  php  7061  marypha1lem  7202  ackbij1lem15  7876  fin23lem38  7991  canthp1lem2  8291  ltexprlem2  8677  suplem1pr  8692  fbfinnfr  17552  ppiltx  20431
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-ne 2461  df-in 3172  df-ss 3179  df-pss 3181
  Copyright terms: Public domain W3C validator