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Theorem bnj98 28899
Description: Technical lemma for bnj150 28908. 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 2791 . . . . . 6  |-  i  e. 
_V
21sucid 4471 . . . . 5  |-  i  e. 
suc  i
3 n0i 3460 . . . . 5  |-  ( i  e.  suc  i  ->  -.  suc  i  =  (/) )
42, 3ax-mp 8 . . . 4  |-  -.  suc  i  =  (/)
5 df-suc 4398 . . . . . 6  |-  suc  i  =  ( i  u. 
{ i } )
6 df-un 3157 . . . . . 6  |-  ( i  u.  { i } )  =  { x  |  ( x  e.  i  \/  x  e. 
{ i } ) }
75, 6eqtri 2303 . . . . 5  |-  suc  i  =  { x  |  ( x  e.  i  \/  x  e.  { i } ) }
8 df1o2 6491 . . . . . . 7  |-  1o  =  { (/) }
97, 8eleq12i 2348 . . . . . 6  |-  ( suc  i  e.  1o  <->  { x  |  ( x  e.  i  \/  x  e. 
{ i } ) }  e.  { (/) } )
10 elsni 3664 . . . . . 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 2327 . . . 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 2610 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 1623    e. wcel 1684   {cab 2269   A.wral 2543    u. cun 3150   (/)c0 3455   {csn 3640   U_ciun 3905   suc csuc 4394   omcom 4656   ` cfv 5255   1oc1o 6472    predc-bnj14 28713
This theorem is referenced by:  bnj150  28908
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-nul 3456  df-sn 3646  df-suc 4398  df-1o 6479
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