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Theorem ackbij1lem1 8056
Description: Lemma for ackbij2 8079. (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 4547 . . . 4  |-  suc  A  =  ( A  u.  { A } )
21ineq2i 3499 . . 3  |-  ( B  i^i  suc  A )  =  ( B  i^i  ( A  u.  { A } ) )
3 indi 3547 . . 3  |-  ( B  i^i  ( A  u.  { A } ) )  =  ( ( B  i^i  A )  u.  ( B  i^i  { A } ) )
42, 3eqtri 2424 . 2  |-  ( B  i^i  suc  A )  =  ( ( B  i^i  A )  u.  ( B  i^i  { A } ) )
5 disjsn 3828 . . . . 5  |-  ( ( B  i^i  { A } )  =  (/)  <->  -.  A  e.  B )
65biimpri 198 . . . 4  |-  ( -.  A  e.  B  -> 
( B  i^i  { A } )  =  (/) )
76uneq2d 3461 . . 3  |-  ( -.  A  e.  B  -> 
( ( B  i^i  A )  u.  ( B  i^i  { A }
) )  =  ( ( B  i^i  A
)  u.  (/) ) )
8 un0 3612 . . 3  |-  ( ( B  i^i  A )  u.  (/) )  =  ( B  i^i  A )
97, 8syl6eq 2452 . 2  |-  ( -.  A  e.  B  -> 
( ( B  i^i  A )  u.  ( B  i^i  { A }
) )  =  ( B  i^i  A ) )
104, 9syl5eq 2448 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 1721    u. cun 3278    i^i cin 3279   (/)c0 3588   {csn 3774   suc csuc 4543
This theorem is referenced by:  ackbij1lem15  8070  ackbij1lem16  8071
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
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 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-nul 3589  df-sn 3780  df-suc 4547
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