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Theorem sorpssi 6530
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 4528 . . 3  |-  ( ( [
C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B [ C.]  C  \/  B  =  C  \/  C [ C.]  B ) )
2 elex 2966 . . . . . 6  |-  ( C  e.  A  ->  C  e.  _V )
32ad2antll 711 . . . . 5  |-  ( ( [
C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  C  e.  _V )
4 brrpssg 6526 . . . . 5  |-  ( C  e.  _V  ->  ( B [ C.]  C  <->  B  C.  C ) )
53, 4syl 16 . . . 4  |-  ( ( [
C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B [ C.]  C  <->  B  C.  C ) )
6 biidd 230 . . . 4  |-  ( ( [
C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B  =  C  <->  B  =  C ) )
7 elex 2966 . . . . . 6  |-  ( B  e.  A  ->  B  e.  _V )
87ad2antrl 710 . . . . 5  |-  ( ( [
C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  B  e.  _V )
9 brrpssg 6526 . . . . 5  |-  ( B  e.  _V  ->  ( C [ C.]  B  <->  C  C.  B ) )
108, 9syl 16 . . . 4  |-  ( ( [
C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( C [ C.]  B  <->  C  C.  B ) )
115, 6, 103orbi123d 1254 . . 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 203 . 2  |-  ( ( [
C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B  C.  C  \/  B  =  C  \/  C  C.  B ) )
13 sspsstri 3451 . 2  |-  ( ( B  C_  C  \/  C  C_  B )  <->  ( B  C.  C  \/  B  =  C  \/  C  C.  B ) )
1412, 13sylibr 205 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 178    \/ wo 359    /\ wa 360    \/ w3o 936    = wceq 1653    e. wcel 1726   _Vcvv 2958    C_ wss 3322    C. wpss 3323   class class class wbr 4214    Or wor 4504   [ C.] crpss 6523
This theorem is referenced by:  sorpssun  6531  sorpssin  6532  sorpssuni  6533  sorpssint  6534  sorpsscmpl  6535  enfin2i  8203  fin1a2lem9  8290  fin1a2lem10  8291  fin1a2lem11  8292  fin1a2lem13  8294
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-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215  df-opab 4269  df-so 4506  df-xp 4886  df-rel 4887  df-rpss 6524
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