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Theorem ackbij1lem2 7863
Description: Lemma for ackbij2 7885. (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 4414 . . . 4  |-  suc  A  =  ( A  u.  { A } )
21ineq2i 3380 . . 3  |-  ( B  i^i  suc  A )  =  ( B  i^i  ( A  u.  { A } ) )
3 indi 3428 . . 3  |-  ( B  i^i  ( A  u.  { A } ) )  =  ( ( B  i^i  A )  u.  ( B  i^i  { A } ) )
4 uncom 3332 . . 3  |-  ( ( B  i^i  A )  u.  ( B  i^i  { A } ) )  =  ( ( B  i^i  { A }
)  u.  ( B  i^i  A ) )
52, 3, 43eqtri 2320 . 2  |-  ( B  i^i  suc  A )  =  ( ( B  i^i  { A }
)  u.  ( B  i^i  A ) )
6 snssi 3775 . . . 4  |-  ( A  e.  B  ->  { A }  C_  B )
7 sseqin2 3401 . . . 4  |-  ( { A }  C_  B  <->  ( B  i^i  { A } )  =  { A } )
86, 7sylib 188 . . 3  |-  ( A  e.  B  ->  ( B  i^i  { A }
)  =  { A } )
98uneq1d 3341 . 2  |-  ( A  e.  B  ->  (
( B  i^i  { A } )  u.  ( B  i^i  A ) )  =  ( { A }  u.  ( B  i^i  A ) ) )
105, 9syl5eq 2340 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 1632    e. wcel 1696    u. cun 3163    i^i cin 3164    C_ wss 3165   {csn 3653   suc csuc 4410
This theorem is referenced by:  ackbij1lem15  7876  ackbij1lem16  7877
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-un 3170  df-in 3172  df-ss 3179  df-sn 3659  df-suc 4414
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