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Theorem sorpssun 6572
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 756 . . 3  |-  ( ( [
C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  C  e.  A )
2 ssequn1 3674 . . . 4  |-  ( B 
C_  C  <->  ( B  u.  C )  =  C )
3 eleq1 2539 . . . 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 755 . . 3  |-  ( ( [
C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  B  e.  A )
7 ssequn2 3677 . . . 4  |-  ( C 
C_  B  <->  ( B  u.  C )  =  B )
8 eleq1 2539 . . . 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 6571 . 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  u.  C )  e.  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    u. cun 3474    C_ wss 3476    Or wor 4799   [ C.] crpss 6564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-so 4801  df-xp 5005  df-rel 5006  df-rpss 6565
This theorem is referenced by:  finsschain  7828  lbsextlem2  17617  lbsextlem3  17618  filssufilg  20239
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