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Theorem sorpssi 6568
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 4823 . . 3  |-  ( ( [
C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B [ C.]  C  \/  B  =  C  \/  C [ C.]  B ) )
2 elex 3122 . . . . . 6  |-  ( C  e.  A  ->  C  e.  _V )
32ad2antll 728 . . . . 5  |-  ( ( [
C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  C  e.  _V )
4 brrpssg 6564 . . . . 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 3122 . . . . . 6  |-  ( B  e.  A  ->  B  e.  _V )
87ad2antrl 727 . . . . 5  |-  ( ( [
C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  B  e.  _V )
9 brrpssg 6564 . . . . 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 3606 . 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 1379    e. wcel 1767   _Vcvv 3113    C_ wss 3476    C. wpss 3477   class class class wbr 4447    Or wor 4799   [ C.] crpss 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-so 4801  df-xp 5005  df-rel 5006  df-rpss 6562
This theorem is referenced by:  sorpssun  6569  sorpssin  6570  sorpssuni  6571  sorpssint  6572  sorpsscmpl  6573  enfin2i  8697  fin1a2lem9  8784  fin1a2lem10  8785  fin1a2lem11  8786  fin1a2lem13  8788
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