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Theorem bnj98 28644
Description: Technical lemma for bnj150 28653. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Assertion
Ref Expression
bnj98  |-  A. i  e.  om  ( suc  i  e.  1o  ->  ( F `  suc  i )  = 
U_ y  e.  ( F `  i ) 
pred ( y ,  A ,  R ) )

Proof of Theorem bnj98
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 vex 2867 . . . . . 6  |-  i  e. 
_V
21sucid 4550 . . . . 5  |-  i  e. 
suc  i
3 n0i 3536 . . . . 5  |-  ( i  e.  suc  i  ->  -.  suc  i  =  (/) )
42, 3ax-mp 8 . . . 4  |-  -.  suc  i  =  (/)
5 df-suc 4477 . . . . . 6  |-  suc  i  =  ( i  u. 
{ i } )
6 df-un 3233 . . . . . 6  |-  ( i  u.  { i } )  =  { x  |  ( x  e.  i  \/  x  e. 
{ i } ) }
75, 6eqtri 2378 . . . . 5  |-  suc  i  =  { x  |  ( x  e.  i  \/  x  e.  { i } ) }
8 df1o2 6575 . . . . . . 7  |-  1o  =  { (/) }
97, 8eleq12i 2423 . . . . . 6  |-  ( suc  i  e.  1o  <->  { x  |  ( x  e.  i  \/  x  e. 
{ i } ) }  e.  { (/) } )
10 elsni 3740 . . . . . 6  |-  ( { x  |  ( x  e.  i  \/  x  e.  { i } ) }  e.  { (/) }  ->  { x  |  ( x  e.  i  \/  x  e.  {
i } ) }  =  (/) )
119, 10sylbi 187 . . . . 5  |-  ( suc  i  e.  1o  ->  { x  |  ( x  e.  i  \/  x  e.  { i } ) }  =  (/) )
127, 11syl5eq 2402 . . . 4  |-  ( suc  i  e.  1o  ->  suc  i  =  (/) )
134, 12mto 167 . . 3  |-  -.  suc  i  e.  1o
1413pm2.21i 123 . 2  |-  ( suc  i  e.  1o  ->  ( F `  suc  i
)  =  U_ y  e.  ( F `  i
)  pred ( y ,  A ,  R ) )
1514rgenw 2686 1  |-  A. i  e.  om  ( suc  i  e.  1o  ->  ( F `  suc  i )  = 
U_ y  e.  ( F `  i ) 
pred ( y ,  A ,  R ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 357    = wceq 1642    e. wcel 1710   {cab 2344   A.wral 2619    u. cun 3226   (/)c0 3531   {csn 3716   U_ciun 3984   suc csuc 4473   omcom 4735   ` cfv 5334   1oc1o 6556    predc-bnj14 28458
This theorem is referenced by:  bnj150  28653
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ral 2624  df-v 2866  df-dif 3231  df-un 3233  df-nul 3532  df-sn 3722  df-suc 4477  df-1o 6563
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