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Theorem sorpssin 6459
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 3270 . . . 4  |-  ( B 
C_  C  <->  ( B  i^i  C )  =  B )
3 eleq1 2440 . . . 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 3481 . . . 4  |-  ( C 
C_  B  <->  ( B  i^i  C )  =  C )
8 eleq1 2440 . . . 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 6457 . 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 1649    e. wcel 1717    i^i cin 3255    C_ wss 3256    Or wor 4436   [ C.] crpss 6450
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-br 4147  df-opab 4201  df-so 4438  df-xp 4817  df-rel 4818  df-rpss 6451
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