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Theorem psssstr 3550
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 3543 . 2  |-  ( B 
C_  C  <->  ( B  C.  C  \/  B  =  C ) )
2 psstr 3548 . . . . 5  |-  ( ( A  C.  B  /\  B  C.  C )  ->  A  C.  C )
32ex 440 . . . 4  |-  ( A 
C.  B  ->  ( B  C.  C  ->  A  C.  C ) )
4 psseq2 3532 . . . . 5  |-  ( B  =  C  ->  ( A  C.  B  <->  A  C.  C
) )
54biimpcd 232 . . . 4  |-  ( A 
C.  B  ->  ( B  =  C  ->  A 
C.  C ) )
63, 5jaod 386 . . 3  |-  ( A 
C.  B  ->  (
( B  C.  C  \/  B  =  C
)  ->  A  C.  C
) )
76imp 435 . 2  |-  ( ( A  C.  B  /\  ( B  C.  C  \/  B  =  C )
)  ->  A  C.  C
)
81, 7sylan2b 482 1  |-  ( ( A  C.  B  /\  B  C_  C )  ->  A  C.  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 374    /\ wa 375    = wceq 1454    C_ wss 3415    C. wpss 3416
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-ne 2634  df-in 3422  df-ss 3429  df-pss 3431
This theorem is referenced by:  psssstrd  3553  suplem1pr  9502  atexch  28082  bj-2upln0  31661  bj-2upln1upl  31662
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