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Theorem sorpssin 6471
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 755 . . 3  |-  ( ( [
C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  B  e.  A )
2 df-ss 3443 . . . 4  |-  ( B 
C_  C  <->  ( B  i^i  C )  =  B )
3 eleq1 2523 . . . 4  |-  ( ( B  i^i  C )  =  B  ->  (
( B  i^i  C
)  e.  A  <->  B  e.  A ) )
42, 3sylbi 195 . . 3  |-  ( B 
C_  C  ->  (
( B  i^i  C
)  e.  A  <->  B  e.  A ) )
51, 4syl5ibrcom 222 . 2  |-  ( ( [
C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B  C_  C  ->  ( B  i^i  C )  e.  A ) )
6 simprr 756 . . 3  |-  ( ( [
C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  C  e.  A )
7 dfss1 3656 . . . 4  |-  ( C 
C_  B  <->  ( B  i^i  C )  =  C )
8 eleq1 2523 . . . 4  |-  ( ( B  i^i  C )  =  C  ->  (
( B  i^i  C
)  e.  A  <->  C  e.  A ) )
97, 8sylbi 195 . . 3  |-  ( C 
C_  B  ->  (
( B  i^i  C
)  e.  A  <->  C  e.  A ) )
106, 9syl5ibrcom 222 . 2  |-  ( ( [
C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( C  C_  B  ->  ( B  i^i  C )  e.  A ) )
11 sorpssi 6469 . 2  |-  ( ( [
C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B  C_  C  \/  C  C_  B ) )
125, 10, 11mpjaod 381 1  |-  ( ( [
C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B  i^i  C )  e.  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    i^i cin 3428    C_ wss 3429    Or wor 4741   [ C.] crpss 6462
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 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pr 4632
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-br 4394  df-opab 4452  df-so 4743  df-xp 4947  df-rel 4948  df-rpss 6463
This theorem is referenced by: (None)
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