MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ackbij1lem2 Structured version   Unicode version

Theorem ackbij1lem2 8652
Description: Lemma for ackbij2 8674. (Contributed by Stefan O'Rear, 18-Nov-2014.)
Assertion
Ref Expression
ackbij1lem2  |-  ( A  e.  B  ->  ( B  i^i  suc  A )  =  ( { A }  u.  ( B  i^i  A ) ) )

Proof of Theorem ackbij1lem2
StepHypRef Expression
1 df-suc 5445 . . . 4  |-  suc  A  =  ( A  u.  { A } )
21ineq2i 3661 . . 3  |-  ( B  i^i  suc  A )  =  ( B  i^i  ( A  u.  { A } ) )
3 indi 3719 . . 3  |-  ( B  i^i  ( A  u.  { A } ) )  =  ( ( B  i^i  A )  u.  ( B  i^i  { A } ) )
4 uncom 3610 . . 3  |-  ( ( B  i^i  A )  u.  ( B  i^i  { A } ) )  =  ( ( B  i^i  { A }
)  u.  ( B  i^i  A ) )
52, 3, 43eqtri 2455 . 2  |-  ( B  i^i  suc  A )  =  ( ( B  i^i  { A }
)  u.  ( B  i^i  A ) )
6 snssi 4141 . . . 4  |-  ( A  e.  B  ->  { A }  C_  B )
7 sseqin2 3681 . . . 4  |-  ( { A }  C_  B  <->  ( B  i^i  { A } )  =  { A } )
86, 7sylib 199 . . 3  |-  ( A  e.  B  ->  ( B  i^i  { A }
)  =  { A } )
98uneq1d 3619 . 2  |-  ( A  e.  B  ->  (
( B  i^i  { A } )  u.  ( B  i^i  A ) )  =  ( { A }  u.  ( B  i^i  A ) ) )
105, 9syl5eq 2475 1  |-  ( A  e.  B  ->  ( B  i^i  suc  A )  =  ( { A }  u.  ( B  i^i  A ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1868    u. cun 3434    i^i cin 3435    C_ wss 3436   {csn 3996   suc csuc 5441
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 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-v 3083  df-un 3441  df-in 3443  df-ss 3450  df-sn 3997  df-suc 5445
This theorem is referenced by:  ackbij1lem15  8665  ackbij1lem16  8666
  Copyright terms: Public domain W3C validator