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Theorem ssnpss 3522
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 3505 . . 3  |-  ( B 
C.  A  <->  ( B  C_  A  /\  -.  A  C_  B ) )
21simprbi 471 . 2  |-  ( B 
C.  A  ->  -.  A  C_  B )
32con2i 124 1  |-  ( A 
C_  B  ->  -.  B  C.  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    C_ wss 3390    C. wpss 3391
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-ne 2643  df-in 3397  df-ss 3404  df-pss 3406
This theorem is referenced by:  sorpssuni  6599  sorpssint  6600  suplem2pr  9496  lsppratlem6  18453  atcvati  28120  finxpreclem3  31855  lsatcvat  32687
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