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Theorem sorpssun 6578
Description: A chain of sets is closed under binary union. (Contributed by Mario Carneiro, 16-May-2015.)
Assertion
Ref Expression
sorpssun  |-  ( ( [ C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A )
)  ->  ( B  u.  C )  e.  A
)

Proof of Theorem sorpssun
StepHypRef Expression
1 simprr 766 . . 3  |-  ( ( [ C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A )
)  ->  C  e.  A )
2 ssequn1 3604 . . . 4  |-  ( B 
C_  C  <->  ( B  u.  C )  =  C )
3 eleq1 2517 . . . 4  |-  ( ( B  u.  C )  =  C  ->  (
( B  u.  C
)  e.  A  <->  C  e.  A ) )
42, 3sylbi 199 . . 3  |-  ( B 
C_  C  ->  (
( B  u.  C
)  e.  A  <->  C  e.  A ) )
51, 4syl5ibrcom 226 . 2  |-  ( ( [ C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A )
)  ->  ( B  C_  C  ->  ( B  u.  C )  e.  A
) )
6 simprl 764 . . 3  |-  ( ( [ C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A )
)  ->  B  e.  A )
7 ssequn2 3607 . . . 4  |-  ( C 
C_  B  <->  ( B  u.  C )  =  B )
8 eleq1 2517 . . . 4  |-  ( ( B  u.  C )  =  B  ->  (
( B  u.  C
)  e.  A  <->  B  e.  A ) )
97, 8sylbi 199 . . 3  |-  ( C 
C_  B  ->  (
( B  u.  C
)  e.  A  <->  B  e.  A ) )
106, 9syl5ibrcom 226 . 2  |-  ( ( [ C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A )
)  ->  ( C  C_  B  ->  ( B  u.  C )  e.  A
) )
11 sorpssi 6577 . 2  |-  ( ( [ C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A )
)  ->  ( B  C_  C  \/  C  C_  B ) )
125, 10, 11mpjaod 383 1  |-  ( ( [ C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A )
)  ->  ( B  u.  C )  e.  A
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1444    e. wcel 1887    u. cun 3402    C_ wss 3404    Or wor 4754   [ C.] crpss 6570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-br 4403  df-opab 4462  df-so 4756  df-xp 4840  df-rel 4841  df-rpss 6571
This theorem is referenced by:  finsschain  7881  lbsextlem2  18382  lbsextlem3  18383  filssufilg  20926
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