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Theorem sorpss 6520
Description: Express strict ordering under proper subsets, i.e. the notion of a chain of sets. (Contributed by Stefan O'Rear, 2-Nov-2014.)
Assertion
Ref Expression
sorpss  |-  ( [ C.]  Or  A  <->  A. x  e.  A  A. y  e.  A  ( x  C_  y  \/  y  C_  x )
)
Distinct variable group:    x, y, A

Proof of Theorem sorpss
StepHypRef Expression
1 porpss 6519 . . 3  |- [ C.]  Po  A
21biantrur 493 . 2  |-  ( A. x  e.  A  A. y  e.  A  (
x [ C.]  y  \/  x  =  y  \/  y [ C.]  x )  <->  ( [ C.]  Po  A  /\  A. x  e.  A  A. y  e.  A  ( x [ C.]  y  \/  x  =  y  \/  y [ C.]  x ) ) )
3 sspsstri 3442 . . . 4  |-  ( ( x  C_  y  \/  y  C_  x )  <->  ( x  C.  y  \/  x  =  y  \/  y  C.  x ) )
4 vex 2952 . . . . . 6  |-  y  e. 
_V
54brrpss 6518 . . . . 5  |-  ( x [
C.]  y  <->  x  C.  y )
6 biid 228 . . . . 5  |-  ( x  =  y  <->  x  =  y )
7 vex 2952 . . . . . 6  |-  x  e. 
_V
87brrpss 6518 . . . . 5  |-  ( y [
C.]  x  <->  y  C.  x )
95, 6, 83orbi123i 1143 . . . 4  |-  ( ( x [ C.]  y  \/  x  =  y  \/  y [ C.]  x )  <->  ( x  C.  y  \/  x  =  y  \/  y  C.  x ) )
103, 9bitr4i 244 . . 3  |-  ( ( x  C_  y  \/  y  C_  x )  <->  ( x [ C.]  y  \/  x  =  y  \/  y [ C.]  x ) )
11102ralbii 2724 . 2  |-  ( A. x  e.  A  A. y  e.  A  (
x  C_  y  \/  y  C_  x )  <->  A. x  e.  A  A. y  e.  A  ( x [ C.]  y  \/  x  =  y  \/  y [ C.]  x ) )
12 df-so 4497 . 2  |-  ( [ C.]  Or  A  <->  ( [ C.]  Po  A  /\  A. x  e.  A  A. y  e.  A  ( x [ C.]  y  \/  x  =  y  \/  y [ C.]  x )
) )
132, 11, 123bitr4ri 270 1  |-  ( [ C.]  Or  A  <->  A. x  e.  A  A. y  e.  A  ( x  C_  y  \/  y  C_  x )
)
Colors of variables: wff set class
Syntax hints:    <-> wb 177    \/ wo 358    /\ wa 359    \/ w3o 935   A.wral 2698    C_ wss 3313    C. wpss 3314   class class class wbr 4205    Po wpo 4494    Or wor 4495   [ C.] crpss 6514
This theorem is referenced by:  sorpsscmpl  6526  enfin2i  8194  fin1a2lem13  8285
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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pr 4396
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-pss 3329  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-br 4206  df-opab 4260  df-po 4496  df-so 4497  df-xp 4877  df-rel 4878  df-rpss 6515
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