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Theorem sorpssi 6585
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 4832 . . 3  |-  ( ( [ C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A )
)  ->  ( B [ C.]  C  \/  B  =  C  \/  C [ C.]  B ) )
2 elex 3118 . . . . . 6  |-  ( C  e.  A  ->  C  e.  _V )
32ad2antll 728 . . . . 5  |-  ( ( [ C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A )
)  ->  C  e.  _V )
4 brrpssg 6581 . . . . 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 237 . . . 4  |-  ( ( [ C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A )
)  ->  ( B  =  C  <->  B  =  C
) )
7 elex 3118 . . . . . 6  |-  ( B  e.  A  ->  B  e.  _V )
87ad2antrl 727 . . . . 5  |-  ( ( [ C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A )
)  ->  B  e.  _V )
9 brrpssg 6581 . . . . 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 1298 . . 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 210 . 2  |-  ( ( [ C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A )
)  ->  ( B  C.  C  \/  B  =  C  \/  C  C.  B ) )
13 sspsstri 3602 . 2  |-  ( ( B  C_  C  \/  C  C_  B )  <->  ( B  C.  C  \/  B  =  C  \/  C  C.  B ) )
1412, 13sylibr 212 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 184    \/ wo 368    /\ wa 369    \/ w3o 972    = wceq 1395    e. wcel 1819   _Vcvv 3109    C_ wss 3471    C. wpss 3472   class class class wbr 4456    Or wor 4808   [ C.] crpss 6578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-so 4810  df-xp 5014  df-rel 5015  df-rpss 6579
This theorem is referenced by:  sorpssun  6586  sorpssin  6587  sorpssuni  6588  sorpssint  6589  sorpsscmpl  6590  enfin2i  8718  fin1a2lem9  8805  fin1a2lem10  8806  fin1a2lem11  8807  fin1a2lem13  8809
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