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Theorem sspsstri 3451
Description: Two ways of stating trichotomy with respect to inclusion. (Contributed by NM, 12-Aug-2004.)
Assertion
Ref Expression
sspsstri  |-  ( ( A  C_  B  \/  B  C_  A )  <->  ( A  C.  B  \/  A  =  B  \/  B  C.  A ) )

Proof of Theorem sspsstri
StepHypRef Expression
1 or32 515 . 2  |-  ( ( ( A  C.  B  \/  B  C.  A )  \/  A  =  B )  <->  ( ( A 
C.  B  \/  A  =  B )  \/  B  C.  A ) )
2 sspss 3448 . . . 4  |-  ( A 
C_  B  <->  ( A  C.  B  \/  A  =  B ) )
3 sspss 3448 . . . . 5  |-  ( B 
C_  A  <->  ( B  C.  A  \/  B  =  A ) )
4 eqcom 2440 . . . . . 6  |-  ( B  =  A  <->  A  =  B )
54orbi2i 507 . . . . 5  |-  ( ( B  C.  A  \/  B  =  A )  <->  ( B  C.  A  \/  A  =  B )
)
63, 5bitri 242 . . . 4  |-  ( B 
C_  A  <->  ( B  C.  A  \/  A  =  B ) )
72, 6orbi12i 509 . . 3  |-  ( ( A  C_  B  \/  B  C_  A )  <->  ( ( A  C.  B  \/  A  =  B )  \/  ( B  C.  A  \/  A  =  B ) ) )
8 orordir 519 . . 3  |-  ( ( ( A  C.  B  \/  B  C.  A )  \/  A  =  B )  <->  ( ( A 
C.  B  \/  A  =  B )  \/  ( B  C.  A  \/  A  =  B ) ) )
97, 8bitr4i 245 . 2  |-  ( ( A  C_  B  \/  B  C_  A )  <->  ( ( A  C.  B  \/  B  C.  A )  \/  A  =  B ) )
10 df-3or 938 . 2  |-  ( ( A  C.  B  \/  A  =  B  \/  B  C.  A )  <->  ( ( A  C.  B  \/  A  =  B )  \/  B  C.  A ) )
111, 9, 103bitr4i 270 1  |-  ( ( A  C_  B  \/  B  C_  A )  <->  ( A  C.  B  \/  A  =  B  \/  B  C.  A ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    \/ wo 359    \/ w3o 936    = wceq 1653    C_ wss 3322    C. wpss 3323
This theorem is referenced by:  ordtri3or  4615  sorpss  6529  sorpssi  6530  funpsstri  25391
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-ne 2603  df-in 3329  df-ss 3336  df-pss 3338
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