MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sorpssi Unicode version

Theorem sorpssi 6299
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 4353 . . 3  |-  ( ( [
C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B [ C.]  C  \/  B  =  C  \/  C [ C.]  B ) )
2 elex 2809 . . . . . 6  |-  ( C  e.  A  ->  C  e.  _V )
32ad2antll 709 . . . . 5  |-  ( ( [
C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  C  e.  _V )
4 brrpssg 6295 . . . . 5  |-  ( C  e.  _V  ->  ( B [ C.]  C  <->  B  C.  C ) )
53, 4syl 15 . . . 4  |-  ( ( [
C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B [ C.]  C  <->  B  C.  C ) )
6 biidd 228 . . . 4  |-  ( ( [
C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B  =  C  <->  B  =  C ) )
7 elex 2809 . . . . . 6  |-  ( B  e.  A  ->  B  e.  _V )
87ad2antrl 708 . . . . 5  |-  ( ( [
C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  B  e.  _V )
9 brrpssg 6295 . . . . 5  |-  ( B  e.  _V  ->  ( C [ C.]  B  <->  C  C.  B ) )
108, 9syl 15 . . . 4  |-  ( ( [
C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( C [ C.]  B  <->  C  C.  B ) )
115, 6, 103orbi123d 1251 . . 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 201 . 2  |-  ( ( [
C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B  C.  C  \/  B  =  C  \/  C  C.  B ) )
13 sspsstri 3291 . 2  |-  ( ( B  C_  C  \/  C  C_  B )  <->  ( B  C.  C  \/  B  =  C  \/  C  C.  B ) )
1412, 13sylibr 203 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 176    \/ wo 357    /\ wa 358    \/ w3o 933    = wceq 1632    e. wcel 1696   _Vcvv 2801    C_ wss 3165    C. wpss 3166   class class class wbr 4039    Or wor 4329   [ C.] crpss 6292
This theorem is referenced by:  sorpssun  6300  sorpssin  6301  sorpssuni  6302  sorpssint  6303  sorpsscmpl  6304  enfin2i  7963  fin1a2lem9  8050  fin1a2lem10  8051  fin1a2lem11  8052  fin1a2lem13  8054
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-so 4331  df-xp 4711  df-rel 4712  df-rpss 6293
  Copyright terms: Public domain W3C validator