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Theorem ackbij1lem2 8036
Description: Lemma for ackbij2 8058. (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 4530 . . . 4  |-  suc  A  =  ( A  u.  { A } )
21ineq2i 3484 . . 3  |-  ( B  i^i  suc  A )  =  ( B  i^i  ( A  u.  { A } ) )
3 indi 3532 . . 3  |-  ( B  i^i  ( A  u.  { A } ) )  =  ( ( B  i^i  A )  u.  ( B  i^i  { A } ) )
4 uncom 3436 . . 3  |-  ( ( B  i^i  A )  u.  ( B  i^i  { A } ) )  =  ( ( B  i^i  { A }
)  u.  ( B  i^i  A ) )
52, 3, 43eqtri 2413 . 2  |-  ( B  i^i  suc  A )  =  ( ( B  i^i  { A }
)  u.  ( B  i^i  A ) )
6 snssi 3887 . . . 4  |-  ( A  e.  B  ->  { A }  C_  B )
7 sseqin2 3505 . . . 4  |-  ( { A }  C_  B  <->  ( B  i^i  { A } )  =  { A } )
86, 7sylib 189 . . 3  |-  ( A  e.  B  ->  ( B  i^i  { A }
)  =  { A } )
98uneq1d 3445 . 2  |-  ( A  e.  B  ->  (
( B  i^i  { A } )  u.  ( B  i^i  A ) )  =  ( { A }  u.  ( B  i^i  A ) ) )
105, 9syl5eq 2433 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 1649    e. wcel 1717    u. cun 3263    i^i cin 3264    C_ wss 3265   {csn 3759   suc csuc 4526
This theorem is referenced by:  ackbij1lem15  8049  ackbij1lem16  8050
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-v 2903  df-un 3270  df-in 3272  df-ss 3279  df-sn 3765  df-suc 4530
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