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Theorem sorpssin 6523
Description: A chain of sets is closed under binary intersection. (Contributed by Mario Carneiro, 16-May-2015.)
Assertion
Ref Expression
sorpssin  |-  ( ( [
C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B  i^i  C )  e.  A )

Proof of Theorem sorpssin
StepHypRef Expression
1 simprl 733 . . 3  |-  ( ( [
C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  B  e.  A )
2 df-ss 3327 . . . 4  |-  ( B 
C_  C  <->  ( B  i^i  C )  =  B )
3 eleq1 2496 . . . 4  |-  ( ( B  i^i  C )  =  B  ->  (
( B  i^i  C
)  e.  A  <->  B  e.  A ) )
42, 3sylbi 188 . . 3  |-  ( B 
C_  C  ->  (
( B  i^i  C
)  e.  A  <->  B  e.  A ) )
51, 4syl5ibrcom 214 . 2  |-  ( ( [
C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B  C_  C  ->  ( B  i^i  C )  e.  A ) )
6 simprr 734 . . 3  |-  ( ( [
C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  C  e.  A )
7 dfss1 3538 . . . 4  |-  ( C 
C_  B  <->  ( B  i^i  C )  =  C )
8 eleq1 2496 . . . 4  |-  ( ( B  i^i  C )  =  C  ->  (
( B  i^i  C
)  e.  A  <->  C  e.  A ) )
97, 8sylbi 188 . . 3  |-  ( C 
C_  B  ->  (
( B  i^i  C
)  e.  A  <->  C  e.  A ) )
106, 9syl5ibrcom 214 . 2  |-  ( ( [
C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( C  C_  B  ->  ( B  i^i  C )  e.  A ) )
11 sorpssi 6521 . 2  |-  ( ( [
C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B  C_  C  \/  C  C_  B ) )
125, 10, 11mpjaod 371 1  |-  ( ( [
C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B  i^i  C )  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    i^i cin 3312    C_ wss 3313    Or wor 4495   [ C.] crpss 6514
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pr 4396
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-pss 3329  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-br 4206  df-opab 4260  df-so 4497  df-xp 4877  df-rel 4878  df-rpss 6515
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