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Theorem bnj98 28944
Description: Technical lemma for bnj150 28953. 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 2919 . . . . . 6  |-  i  e. 
_V
21sucid 4620 . . . . 5  |-  i  e. 
suc  i
3 n0i 3593 . . . . 5  |-  ( i  e.  suc  i  ->  -.  suc  i  =  (/) )
42, 3ax-mp 8 . . . 4  |-  -.  suc  i  =  (/)
5 df-suc 4547 . . . . . 6  |-  suc  i  =  ( i  u. 
{ i } )
6 df-un 3285 . . . . . 6  |-  ( i  u.  { i } )  =  { x  |  ( x  e.  i  \/  x  e. 
{ i } ) }
75, 6eqtri 2424 . . . . 5  |-  suc  i  =  { x  |  ( x  e.  i  \/  x  e.  { i } ) }
8 df1o2 6695 . . . . . . 7  |-  1o  =  { (/) }
97, 8eleq12i 2469 . . . . . 6  |-  ( suc  i  e.  1o  <->  { x  |  ( x  e.  i  \/  x  e. 
{ i } ) }  e.  { (/) } )
10 elsni 3798 . . . . . 6  |-  ( { x  |  ( x  e.  i  \/  x  e.  { i } ) }  e.  { (/) }  ->  { x  |  ( x  e.  i  \/  x  e.  {
i } ) }  =  (/) )
119, 10sylbi 188 . . . . 5  |-  ( suc  i  e.  1o  ->  { x  |  ( x  e.  i  \/  x  e.  { i } ) }  =  (/) )
127, 11syl5eq 2448 . . . 4  |-  ( suc  i  e.  1o  ->  suc  i  =  (/) )
134, 12mto 169 . . 3  |-  -.  suc  i  e.  1o
1413pm2.21i 125 . 2  |-  ( suc  i  e.  1o  ->  ( F `  suc  i
)  =  U_ y  e.  ( F `  i
)  pred ( y ,  A ,  R ) )
1514rgenw 2733 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 358    = wceq 1649    e. wcel 1721   {cab 2390   A.wral 2666    u. cun 3278   (/)c0 3588   {csn 3774   U_ciun 4053   suc csuc 4543   omcom 4804   ` cfv 5413   1oc1o 6676    predc-bnj14 28758
This theorem is referenced by:  bnj150  28953
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-v 2918  df-dif 3283  df-un 3285  df-nul 3589  df-sn 3780  df-suc 4547  df-1o 6683
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