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Theorem sorpssun 6560
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 755 . . 3  |-  ( ( [ C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A )
)  ->  C  e.  A )
2 ssequn1 3660 . . . 4  |-  ( B 
C_  C  <->  ( B  u.  C )  =  C )
3 eleq1 2526 . . . 4  |-  ( ( B  u.  C )  =  C  ->  (
( B  u.  C
)  e.  A  <->  C  e.  A ) )
42, 3sylbi 195 . . 3  |-  ( B 
C_  C  ->  (
( B  u.  C
)  e.  A  <->  C  e.  A ) )
51, 4syl5ibrcom 222 . 2  |-  ( ( [ C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A )
)  ->  ( B  C_  C  ->  ( B  u.  C )  e.  A
) )
6 simprl 754 . . 3  |-  ( ( [ C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A )
)  ->  B  e.  A )
7 ssequn2 3663 . . . 4  |-  ( C 
C_  B  <->  ( B  u.  C )  =  B )
8 eleq1 2526 . . . 4  |-  ( ( B  u.  C )  =  B  ->  (
( B  u.  C
)  e.  A  <->  B  e.  A ) )
97, 8sylbi 195 . . 3  |-  ( C 
C_  B  ->  (
( B  u.  C
)  e.  A  <->  B  e.  A ) )
106, 9syl5ibrcom 222 . 2  |-  ( ( [ C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A )
)  ->  ( C  C_  B  ->  ( B  u.  C )  e.  A
) )
11 sorpssi 6559 . 2  |-  ( ( [ C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A )
)  ->  ( B  C_  C  \/  C  C_  B ) )
125, 10, 11mpjaod 379 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 184    /\ wa 367    = wceq 1398    e. wcel 1823    u. cun 3459    C_ wss 3461    Or wor 4788   [ C.] crpss 6552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-so 4790  df-xp 4994  df-rel 4995  df-rpss 6553
This theorem is referenced by:  finsschain  7819  lbsextlem2  18000  lbsextlem3  18001  filssufilg  20578
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