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Theorem psssstr 3413
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 3406 . 2  |-  ( B 
C_  C  <->  ( B  C.  C  \/  B  =  C ) )
2 psstr 3411 . . . . 5  |-  ( ( A  C.  B  /\  B  C.  C )  ->  A  C.  C )
32ex 424 . . . 4  |-  ( A 
C.  B  ->  ( B  C.  C  ->  A  C.  C ) )
4 psseq2 3395 . . . . 5  |-  ( B  =  C  ->  ( A  C.  B  <->  A  C.  C ) )
54biimpcd 216 . . . 4  |-  ( A 
C.  B  ->  ( B  =  C  ->  A 
C.  C ) )
63, 5jaod 370 . . 3  |-  ( A 
C.  B  ->  (
( B  C.  C  \/  B  =  C
)  ->  A  C.  C ) )
76imp 419 . 2  |-  ( ( A  C.  B  /\  ( B  C.  C  \/  B  =  C )
)  ->  A  C.  C )
81, 7sylan2b 462 1  |-  ( ( A  C.  B  /\  B  C_  C )  ->  A  C.  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    = wceq 1649    C_ wss 3280    C. wpss 3281
This theorem is referenced by:  psssstrd  3416  suplem1pr  8885  atexch  23837
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-ne 2569  df-in 3287  df-ss 3294  df-pss 3296
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