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Theorem ackbij1lem1 8131
Description: Lemma for ackbij2 8154. (Contributed by Stefan O'Rear, 18-Nov-2014.)
Assertion
Ref Expression
ackbij1lem1  |-  ( -.  A  e.  B  -> 
( B  i^i  suc  A )  =  ( B  i^i  A ) )

Proof of Theorem ackbij1lem1
StepHypRef Expression
1 df-suc 4616 . . . 4  |-  suc  A  =  ( A  u.  { A } )
21ineq2i 3525 . . 3  |-  ( B  i^i  suc  A )  =  ( B  i^i  ( A  u.  { A } ) )
3 indi 3572 . . 3  |-  ( B  i^i  ( A  u.  { A } ) )  =  ( ( B  i^i  A )  u.  ( B  i^i  { A } ) )
42, 3eqtri 2462 . 2  |-  ( B  i^i  suc  A )  =  ( ( B  i^i  A )  u.  ( B  i^i  { A } ) )
5 disjsn 3892 . . . . 5  |-  ( ( B  i^i  { A } )  =  (/)  <->  -.  A  e.  B )
65biimpri 199 . . . 4  |-  ( -.  A  e.  B  -> 
( B  i^i  { A } )  =  (/) )
76uneq2d 3487 . . 3  |-  ( -.  A  e.  B  -> 
( ( B  i^i  A )  u.  ( B  i^i  { A }
) )  =  ( ( B  i^i  A
)  u.  (/) ) )
8 un0 3637 . . 3  |-  ( ( B  i^i  A )  u.  (/) )  =  ( B  i^i  A )
97, 8syl6eq 2490 . 2  |-  ( -.  A  e.  B  -> 
( ( B  i^i  A )  u.  ( B  i^i  { A }
) )  =  ( B  i^i  A ) )
104, 9syl5eq 2486 1  |-  ( -.  A  e.  B  -> 
( B  i^i  suc  A )  =  ( B  i^i  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1653    e. wcel 1727    u. cun 3304    i^i cin 3305   (/)c0 3613   {csn 3838   suc csuc 4612
This theorem is referenced by:  ackbij1lem15  8145  ackbij1lem16  8146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ral 2716  df-v 2964  df-dif 3309  df-un 3311  df-in 3313  df-nul 3614  df-sn 3844  df-suc 4616
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