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Theorem ssnpss 3442
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 3425 . . 3  |-  ( B 
C.  A  <->  ( B  C_  A  /\  -.  A  C_  B ) )
21simprbi 451 . 2  |-  ( B 
C.  A  ->  -.  A  C_  B )
32con2i 114 1  |-  ( A 
C_  B  ->  -.  B  C.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    C_ wss 3312    C. wpss 3313
This theorem is referenced by:  sorpssuni  6523  sorpssint  6524  suplem2pr  8922  lsppratlem6  16216  atcvati  23881  lsatcvat  29785
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-ne 2600  df-in 3319  df-ss 3326  df-pss 3328
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