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Theorem sorpss 6282
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 6281 . . 3  |- [ C.]  Po  A
21biantrur 492 . 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 3278 . . . 4  |-  ( ( x  C_  y  \/  y  C_  x )  <->  ( x  C.  y  \/  x  =  y  \/  y  C.  x ) )
4 vex 2791 . . . . . 6  |-  y  e. 
_V
54brrpss 6280 . . . . 5  |-  ( x [
C.]  y  <->  x  C.  y )
6 biid 227 . . . . 5  |-  ( x  =  y  <->  x  =  y )
7 vex 2791 . . . . . 6  |-  x  e. 
_V
87brrpss 6280 . . . . 5  |-  ( y [
C.]  x  <->  y  C.  x )
95, 6, 83orbi123i 1141 . . . 4  |-  ( ( x [ C.]  y  \/  x  =  y  \/  y [ C.]  x )  <->  ( x  C.  y  \/  x  =  y  \/  y  C.  x ) )
103, 9bitr4i 243 . . 3  |-  ( ( x  C_  y  \/  y  C_  x )  <->  ( x [ C.]  y  \/  x  =  y  \/  y [ C.]  x ) )
11102ralbii 2569 . 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 4315 . 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 269 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 176    \/ wo 357    /\ wa 358    \/ w3o 933   A.wral 2543    C_ wss 3152    C. wpss 3153   class class class wbr 4023    Po wpo 4312    Or wor 4313   [ C.] crpss 6276
This theorem is referenced by:  sorpsscmpl  6288  enfin2i  7947  fin1a2lem13  8038
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-rpss 6277
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