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

Theorem psssstr 3455
Description: Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.)
Assertion
Ref Expression
psssstr  |-  ( ( A  C.  B  /\  B  C_  C )  ->  A  C.  C )

Proof of Theorem psssstr
StepHypRef Expression
1 sspss 3448 . 2  |-  ( B 
C_  C  <->  ( B  C.  C  \/  B  =  C ) )
2 psstr 3453 . . . . 5  |-  ( ( A  C.  B  /\  B  C.  C )  ->  A  C.  C )
32ex 425 . . . 4  |-  ( A 
C.  B  ->  ( B  C.  C  ->  A  C.  C ) )
4 psseq2 3437 . . . . 5  |-  ( B  =  C  ->  ( A  C.  B  <->  A  C.  C ) )
54biimpcd 217 . . . 4  |-  ( A 
C.  B  ->  ( B  =  C  ->  A 
C.  C ) )
63, 5jaod 371 . . 3  |-  ( A 
C.  B  ->  (
( B  C.  C  \/  B  =  C
)  ->  A  C.  C ) )
76imp 420 . 2  |-  ( ( A  C.  B  /\  ( B  C.  C  \/  B  =  C )
)  ->  A  C.  C )
81, 7sylan2b 463 1  |-  ( ( A  C.  B  /\  B  C_  C )  ->  A  C.  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 359    /\ wa 360    = wceq 1653    C_ wss 3322    C. wpss 3323
This theorem is referenced by:  psssstrd  3458  suplem1pr  8934  atexch  23889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-ne 2603  df-in 3329  df-ss 3336  df-pss 3338
  Copyright terms: Public domain W3C validator