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Theorem sorpssi 6477
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 4773 . . 3  |-  ( ( [
C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B [ C.]  C  \/  B  =  C  \/  C [ C.]  B ) )
2 elex 3087 . . . . . 6  |-  ( C  e.  A  ->  C  e.  _V )
32ad2antll 728 . . . . 5  |-  ( ( [
C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  C  e.  _V )
4 brrpssg 6473 . . . . 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 3087 . . . . . 6  |-  ( B  e.  A  ->  B  e.  _V )
87ad2antrl 727 . . . . 5  |-  ( ( [
C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  B  e.  _V )
9 brrpssg 6473 . . . . 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 1289 . . 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 3567 . 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 964    = wceq 1370    e. wcel 1758   _Vcvv 3078    C_ wss 3437    C. wpss 3438   class class class wbr 4401    Or wor 4749   [ C.] crpss 6470
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-br 4402  df-opab 4460  df-so 4751  df-xp 4955  df-rel 4956  df-rpss 6471
This theorem is referenced by:  sorpssun  6478  sorpssin  6479  sorpssuni  6480  sorpssint  6481  sorpsscmpl  6482  enfin2i  8602  fin1a2lem9  8689  fin1a2lem10  8690  fin1a2lem11  8691  fin1a2lem13  8693
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