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Theorem sorpssi 6283
Description: Property of a chain of sets. (Contributed by Stefan O'Rear, 2-Nov-2014.)
Assertion
Ref Expression
sorpssi  |-  ( ( [
C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B  C_  C  \/  C  C_  B ) )

Proof of Theorem sorpssi
StepHypRef Expression
1 solin 4337 . . 3  |-  ( ( [
C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B [ C.]  C  \/  B  =  C  \/  C [ C.]  B ) )
2 elex 2796 . . . . . 6  |-  ( C  e.  A  ->  C  e.  _V )
32ad2antll 709 . . . . 5  |-  ( ( [
C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  C  e.  _V )
4 brrpssg 6279 . . . . 5  |-  ( C  e.  _V  ->  ( B [ C.]  C  <->  B  C.  C ) )
53, 4syl 15 . . . 4  |-  ( ( [
C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B [ C.]  C  <->  B  C.  C ) )
6 biidd 228 . . . 4  |-  ( ( [
C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B  =  C  <->  B  =  C ) )
7 elex 2796 . . . . . 6  |-  ( B  e.  A  ->  B  e.  _V )
87ad2antrl 708 . . . . 5  |-  ( ( [
C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  B  e.  _V )
9 brrpssg 6279 . . . . 5  |-  ( B  e.  _V  ->  ( C [ C.]  B  <->  C  C.  B ) )
108, 9syl 15 . . . 4  |-  ( ( [
C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( C [ C.]  B  <->  C  C.  B ) )
115, 6, 103orbi123d 1251 . . 3  |-  ( ( [
C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  (
( B [ C.]  C  \/  B  =  C  \/  C [ C.]  B )  <-> 
( B  C.  C  \/  B  =  C  \/  C  C.  B ) ) )
121, 11mpbid 201 . 2  |-  ( ( [
C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B  C.  C  \/  B  =  C  \/  C  C.  B ) )
13 sspsstri 3278 . 2  |-  ( ( B  C_  C  \/  C  C_  B )  <->  ( B  C.  C  \/  B  =  C  \/  C  C.  B ) )
1412, 13sylibr 203 1  |-  ( ( [
C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B  C_  C  \/  C  C_  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    \/ w3o 933    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152    C. wpss 3153   class class class wbr 4023    Or wor 4313   [ C.] crpss 6276
This theorem is referenced by:  sorpssun  6284  sorpssin  6285  sorpssuni  6286  sorpssint  6287  sorpsscmpl  6288  enfin2i  7947  fin1a2lem9  8034  fin1a2lem10  8035  fin1a2lem11  8036  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-so 4315  df-xp 4695  df-rel 4696  df-rpss 6277
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