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

Theorem sspsstr 3281
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 3275 . 2  |-  ( A 
C_  B  <->  ( A  C.  B  \/  A  =  B ) )
2 psstr 3280 . . . . 5  |-  ( ( A  C.  B  /\  B  C.  C )  ->  A  C.  C )
32ex 423 . . . 4  |-  ( A 
C.  B  ->  ( B  C.  C  ->  A  C.  C ) )
4 psseq1 3263 . . . . 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 1623    C_ wss 3152    C. wpss 3153
This theorem is referenced by:  sspsstrd  3284  ordtr2  4436  php  7045  marypha1lem  7186  ackbij1lem15  7860  fin23lem38  7975  canthp1lem2  8275  ltexprlem2  8661  suplem1pr  8676  fbfinnfr  17536  ppiltx  20415
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-ne 2448  df-in 3159  df-ss 3166  df-pss 3168
  Copyright terms: Public domain W3C validator