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Theorem ackbij1lem1 8035
Description: Lemma for ackbij2 8058. (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 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 } ) )
42, 3eqtri 2409 . 2  |-  ( B  i^i  suc  A )  =  ( ( B  i^i  A )  u.  ( B  i^i  { A } ) )
5 disjsn 3813 . . . . 5  |-  ( ( B  i^i  { A } )  =  (/)  <->  -.  A  e.  B )
65biimpri 198 . . . 4  |-  ( -.  A  e.  B  -> 
( B  i^i  { A } )  =  (/) )
76uneq2d 3446 . . 3  |-  ( -.  A  e.  B  -> 
( ( B  i^i  A )  u.  ( B  i^i  { A }
) )  =  ( ( B  i^i  A
)  u.  (/) ) )
8 un0 3597 . . 3  |-  ( ( B  i^i  A )  u.  (/) )  =  ( B  i^i  A )
97, 8syl6eq 2437 . 2  |-  ( -.  A  e.  B  -> 
( ( B  i^i  A )  u.  ( B  i^i  { A }
) )  =  ( B  i^i  A ) )
104, 9syl5eq 2433 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 1649    e. wcel 1717    u. cun 3263    i^i cin 3264   (/)c0 3573   {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-ral 2656  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-nul 3574  df-sn 3765  df-suc 4530
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