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Theorem sorpssi 6603
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 4796 . . 3  |-  ( ( [ C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A )
)  ->  ( B [ C.]  C  \/  B  =  C  \/  C [ C.]  B ) )
2 elex 3065 . . . . . 6  |-  ( C  e.  A  ->  C  e.  _V )
32ad2antll 740 . . . . 5  |-  ( ( [ C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A )
)  ->  C  e.  _V )
4 brrpssg 6599 . . . . 5  |-  ( C  e.  _V  ->  ( B [ C.]  C  <->  B  C.  C
) )
53, 4syl 17 . . . 4  |-  ( ( [ C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A )
)  ->  ( B [ C.]  C  <->  B  C.  C ) )
6 biidd 245 . . . 4  |-  ( ( [ C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A )
)  ->  ( B  =  C  <->  B  =  C
) )
7 elex 3065 . . . . . 6  |-  ( B  e.  A  ->  B  e.  _V )
87ad2antrl 739 . . . . 5  |-  ( ( [ C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A )
)  ->  B  e.  _V )
9 brrpssg 6599 . . . . 5  |-  ( B  e.  _V  ->  ( C [ C.]  B  <->  C  C.  B
) )
108, 9syl 17 . . . 4  |-  ( ( [ C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A )
)  ->  ( C [ C.]  B  <->  C  C.  B ) )
115, 6, 103orbi123d 1347 . . 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 215 . 2  |-  ( ( [ C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A )
)  ->  ( B  C.  C  \/  B  =  C  \/  C  C.  B ) )
13 sspsstri 3546 . 2  |-  ( ( B  C_  C  \/  C  C_  B )  <->  ( B  C.  C  \/  B  =  C  \/  C  C.  B ) )
1412, 13sylibr 217 1  |-  ( ( [ C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A )
)  ->  ( B  C_  C  \/  C  C_  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    \/ wo 374    /\ wa 375    \/ w3o 990    = wceq 1454    e. wcel 1897   _Vcvv 3056    C_ wss 3415    C. wpss 3416   class class class wbr 4415    Or wor 4772   [ C.] crpss 6596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-sn 3980  df-pr 3982  df-op 3986  df-br 4416  df-opab 4475  df-so 4774  df-xp 4858  df-rel 4859  df-rpss 6597
This theorem is referenced by:  sorpssun  6604  sorpssin  6605  sorpssuni  6606  sorpssint  6607  sorpsscmpl  6608  enfin2i  8776  fin1a2lem9  8863  fin1a2lem10  8864  fin1a2lem11  8865  fin1a2lem13  8867
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