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Theorem sorpssun 6530
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 735 . . 3  |-  ( ( [
C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  C  e.  A )
2 ssequn1 3518 . . . 4  |-  ( B 
C_  C  <->  ( B  u.  C )  =  C )
3 eleq1 2497 . . . 4  |-  ( ( B  u.  C )  =  C  ->  (
( B  u.  C
)  e.  A  <->  C  e.  A ) )
42, 3sylbi 189 . . 3  |-  ( B 
C_  C  ->  (
( B  u.  C
)  e.  A  <->  C  e.  A ) )
51, 4syl5ibrcom 215 . 2  |-  ( ( [
C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B  C_  C  ->  ( B  u.  C )  e.  A ) )
6 simprl 734 . . 3  |-  ( ( [
C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  B  e.  A )
7 ssequn2 3521 . . . 4  |-  ( C 
C_  B  <->  ( B  u.  C )  =  B )
8 eleq1 2497 . . . 4  |-  ( ( B  u.  C )  =  B  ->  (
( B  u.  C
)  e.  A  <->  B  e.  A ) )
97, 8sylbi 189 . . 3  |-  ( C 
C_  B  ->  (
( B  u.  C
)  e.  A  <->  B  e.  A ) )
106, 9syl5ibrcom 215 . 2  |-  ( ( [
C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( C  C_  B  ->  ( B  u.  C )  e.  A ) )
11 sorpssi 6529 . 2  |-  ( ( [
C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B  C_  C  \/  C  C_  B ) )
125, 10, 11mpjaod 372 1  |-  ( ( [
C.]  Or  A  /\  ( B  e.  A  /\  C  e.  A
) )  ->  ( B  u.  C )  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    u. cun 3319    C_ wss 3321    Or wor 4503   [ C.] crpss 6522
This theorem is referenced by:  finsschain  7414  lbsextlem2  16232  lbsextlem3  16233  filssufilg  17944
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-br 4214  df-opab 4268  df-so 4505  df-xp 4885  df-rel 4886  df-rpss 6523
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