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Theorem ackbij1lem1 8646
Description: Lemma for ackbij2 8669. (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 5440 . . . 4  |-  suc  A  =  ( A  u.  { A } )
21ineq2i 3658 . . 3  |-  ( B  i^i  suc  A )  =  ( B  i^i  ( A  u.  { A } ) )
3 indi 3716 . . 3  |-  ( B  i^i  ( A  u.  { A } ) )  =  ( ( B  i^i  A )  u.  ( B  i^i  { A } ) )
42, 3eqtri 2449 . 2  |-  ( B  i^i  suc  A )  =  ( ( B  i^i  A )  u.  ( B  i^i  { A } ) )
5 disjsn 4054 . . . . 5  |-  ( ( B  i^i  { A } )  =  (/)  <->  -.  A  e.  B )
65biimpri 209 . . . 4  |-  ( -.  A  e.  B  -> 
( B  i^i  { A } )  =  (/) )
76uneq2d 3617 . . 3  |-  ( -.  A  e.  B  -> 
( ( B  i^i  A )  u.  ( B  i^i  { A }
) )  =  ( ( B  i^i  A
)  u.  (/) ) )
8 un0 3784 . . 3  |-  ( ( B  i^i  A )  u.  (/) )  =  ( B  i^i  A )
97, 8syl6eq 2477 . 2  |-  ( -.  A  e.  B  -> 
( ( B  i^i  A )  u.  ( B  i^i  { A }
) )  =  ( B  i^i  A ) )
104, 9syl5eq 2473 1  |-  ( -.  A  e.  B  -> 
( B  i^i  suc  A )  =  ( B  i^i  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1437    e. wcel 1867    u. cun 3431    i^i cin 3432   (/)c0 3758   {csn 3993   suc csuc 5436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ral 2778  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-nul 3759  df-sn 3994  df-suc 5440
This theorem is referenced by:  ackbij1lem15  8660  ackbij1lem16  8661
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