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Theorem bnj98 31860
Description: Technical lemma for bnj150 31869. 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 2975 . . . . . 6  |-  i  e. 
_V
21sucid 4798 . . . . 5  |-  i  e. 
suc  i
3 n0i 3642 . . . . 5  |-  ( i  e.  suc  i  ->  -.  suc  i  =  (/) )
42, 3ax-mp 5 . . . 4  |-  -.  suc  i  =  (/)
5 df-suc 4725 . . . . . 6  |-  suc  i  =  ( i  u. 
{ i } )
6 df-un 3333 . . . . . 6  |-  ( i  u.  { i } )  =  { x  |  ( x  e.  i  \/  x  e. 
{ i } ) }
75, 6eqtri 2463 . . . . 5  |-  suc  i  =  { x  |  ( x  e.  i  \/  x  e.  { i } ) }
8 df1o2 6932 . . . . . . 7  |-  1o  =  { (/) }
97, 8eleq12i 2508 . . . . . 6  |-  ( suc  i  e.  1o  <->  { x  |  ( x  e.  i  \/  x  e. 
{ i } ) }  e.  { (/) } )
10 elsni 3902 . . . . . 6  |-  ( { x  |  ( x  e.  i  \/  x  e.  { i } ) }  e.  { (/) }  ->  { x  |  ( x  e.  i  \/  x  e.  {
i } ) }  =  (/) )
119, 10sylbi 195 . . . . 5  |-  ( suc  i  e.  1o  ->  { x  |  ( x  e.  i  \/  x  e.  { i } ) }  =  (/) )
127, 11syl5eq 2487 . . . 4  |-  ( suc  i  e.  1o  ->  suc  i  =  (/) )
134, 12mto 176 . . 3  |-  -.  suc  i  e.  1o
1413pm2.21i 131 . 2  |-  ( suc  i  e.  1o  ->  ( F `  suc  i
)  =  U_ y  e.  ( F `  i
)  pred ( y ,  A ,  R ) )
1514rgenw 2783 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    = wceq 1369    e. wcel 1756   {cab 2429   A.wral 2715    u. cun 3326   (/)c0 3637   {csn 3877   U_ciun 4171   suc csuc 4721   ` cfv 5418   omcom 6476   1oc1o 6913    predc-bnj14 31676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ral 2720  df-v 2974  df-dif 3331  df-un 3333  df-nul 3638  df-sn 3878  df-suc 4725  df-1o 6920
This theorem is referenced by:  bnj150  31869
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