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Theorem ssnpss 3047
Description: Partial trichotomy law for subclasses. (Contributed by NM, 16-May-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
ssnpss  |-  ( A 
C_  B  ->  -.  B  C.  A )

Proof of Theorem ssnpss
StepHypRef Expression
1 dfpss3 3030 . . 3  |-  ( B 
C.  A  <->  ( B  C_  A  /\  -.  A  C_  B ) )
21simprbi 260 . 2  |-  ( B 
C.  A  ->  -.  A  C_  B )
32con2i 557 1  |-  ( A 
C_  B  ->  -.  B  C.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    C_ wss 2917    C. wpss 2918
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-ne 2206  df-in 2924  df-ss 2931  df-pss 2933
This theorem is referenced by:  sspssn  3048
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