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Theorem ackbij1lem2 7847
Description: Lemma for ackbij2 7869. (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 4398 . . . 4  |-  suc  A  =  ( A  u.  { A } )
21ineq2i 3367 . . 3  |-  ( B  i^i  suc  A )  =  ( B  i^i  ( A  u.  { A } ) )
3 indi 3415 . . 3  |-  ( B  i^i  ( A  u.  { A } ) )  =  ( ( B  i^i  A )  u.  ( B  i^i  { A } ) )
4 uncom 3319 . . 3  |-  ( ( B  i^i  A )  u.  ( B  i^i  { A } ) )  =  ( ( B  i^i  { A }
)  u.  ( B  i^i  A ) )
52, 3, 43eqtri 2307 . 2  |-  ( B  i^i  suc  A )  =  ( ( B  i^i  { A }
)  u.  ( B  i^i  A ) )
6 snssi 3759 . . . 4  |-  ( A  e.  B  ->  { A }  C_  B )
7 sseqin2 3388 . . . 4  |-  ( { A }  C_  B  <->  ( B  i^i  { A } )  =  { A } )
86, 7sylib 188 . . 3  |-  ( A  e.  B  ->  ( B  i^i  { A }
)  =  { A } )
98uneq1d 3328 . 2  |-  ( A  e.  B  ->  (
( B  i^i  { A } )  u.  ( B  i^i  A ) )  =  ( { A }  u.  ( B  i^i  A ) ) )
105, 9syl5eq 2327 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 1623    e. wcel 1684    u. cun 3150    i^i cin 3151    C_ wss 3152   {csn 3640   suc csuc 4394
This theorem is referenced by:  ackbij1lem15  7860  ackbij1lem16  7861
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-un 3157  df-in 3159  df-ss 3166  df-sn 3646  df-suc 4398
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