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Theorem ackbij1lem2 8091
Description: Lemma for ackbij2 8113. (Contributed by Stefan O'Rear, 18-Nov-2014.)
Assertion
Ref Expression
ackbij1lem2  |-  ( A  e.  B  ->  ( B  i^i  suc  A )  =  ( { A }  u.  ( B  i^i  A ) ) )

Proof of Theorem ackbij1lem2
StepHypRef Expression
1 df-suc 4579 . . . 4  |-  suc  A  =  ( A  u.  { A } )
21ineq2i 3531 . . 3  |-  ( B  i^i  suc  A )  =  ( B  i^i  ( A  u.  { A } ) )
3 indi 3579 . . 3  |-  ( B  i^i  ( A  u.  { A } ) )  =  ( ( B  i^i  A )  u.  ( B  i^i  { A } ) )
4 uncom 3483 . . 3  |-  ( ( B  i^i  A )  u.  ( B  i^i  { A } ) )  =  ( ( B  i^i  { A }
)  u.  ( B  i^i  A ) )
52, 3, 43eqtri 2459 . 2  |-  ( B  i^i  suc  A )  =  ( ( B  i^i  { A }
)  u.  ( B  i^i  A ) )
6 snssi 3934 . . . 4  |-  ( A  e.  B  ->  { A }  C_  B )
7 sseqin2 3552 . . . 4  |-  ( { A }  C_  B  <->  ( B  i^i  { A } )  =  { A } )
86, 7sylib 189 . . 3  |-  ( A  e.  B  ->  ( B  i^i  { A }
)  =  { A } )
98uneq1d 3492 . 2  |-  ( A  e.  B  ->  (
( B  i^i  { A } )  u.  ( B  i^i  A ) )  =  ( { A }  u.  ( B  i^i  A ) ) )
105, 9syl5eq 2479 1  |-  ( A  e.  B  ->  ( B  i^i  suc  A )  =  ( { A }  u.  ( B  i^i  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725    u. cun 3310    i^i cin 3311    C_ wss 3312   {csn 3806   suc csuc 4575
This theorem is referenced by:  ackbij1lem15  8104  ackbij1lem16  8105
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-un 3317  df-in 3319  df-ss 3326  df-sn 3812  df-suc 4579
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