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Theorem ackbij1lem1 7846
Description: Lemma for ackbij2 7869. (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 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 } ) )
42, 3eqtri 2303 . 2  |-  ( B  i^i  suc  A )  =  ( ( B  i^i  A )  u.  ( B  i^i  { A } ) )
5 disjsn 3693 . . . . 5  |-  ( ( B  i^i  { A } )  =  (/)  <->  -.  A  e.  B )
65biimpri 197 . . . 4  |-  ( -.  A  e.  B  -> 
( B  i^i  { A } )  =  (/) )
76uneq2d 3329 . . 3  |-  ( -.  A  e.  B  -> 
( ( B  i^i  A )  u.  ( B  i^i  { A }
) )  =  ( ( B  i^i  A
)  u.  (/) ) )
8 un0 3479 . . 3  |-  ( ( B  i^i  A )  u.  (/) )  =  ( B  i^i  A )
97, 8syl6eq 2331 . 2  |-  ( -.  A  e.  B  -> 
( ( B  i^i  A )  u.  ( B  i^i  { A }
) )  =  ( B  i^i  A ) )
104, 9syl5eq 2327 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 1623    e. wcel 1684    u. cun 3150    i^i cin 3151   (/)c0 3455   {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-ral 2548  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-nul 3456  df-sn 3646  df-suc 4398
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