MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sorpssun Structured version   Unicode version

Theorem sorpssun 6379
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 3538 . . . 4  |-  ( B 
C_  C  <->  ( B  u.  C )  =  C )
3 eleq1 2503 . . . 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 3541 . . . 4  |-  ( C 
C_  B  <->  ( B  u.  C )  =  B )
8 eleq1 2503 . . . 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 6378 . 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 1369    e. wcel 1756    u. cun 3338    C_ wss 3340    Or wor 4652   [ C.] crpss 6371
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pr 4543
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-br 4305  df-opab 4363  df-so 4654  df-xp 4858  df-rel 4859  df-rpss 6372
This theorem is referenced by:  finsschain  7630  lbsextlem2  17252  lbsextlem3  17253  filssufilg  19496
  Copyright terms: Public domain W3C validator