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Theorem bnj98 31694
Description: Technical lemma for bnj150 31703. 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 2973 . . . . . 6  |-  i  e. 
_V
21sucid 4794 . . . . 5  |-  i  e. 
suc  i
3 n0i 3639 . . . . 5  |-  ( i  e.  suc  i  ->  -.  suc  i  =  (/) )
42, 3ax-mp 5 . . . 4  |-  -.  suc  i  =  (/)
5 df-suc 4721 . . . . . 6  |-  suc  i  =  ( i  u. 
{ i } )
6 df-un 3330 . . . . . 6  |-  ( i  u.  { i } )  =  { x  |  ( x  e.  i  \/  x  e. 
{ i } ) }
75, 6eqtri 2461 . . . . 5  |-  suc  i  =  { x  |  ( x  e.  i  \/  x  e.  { i } ) }
8 df1o2 6928 . . . . . . 7  |-  1o  =  { (/) }
97, 8eleq12i 2506 . . . . . 6  |-  ( suc  i  e.  1o  <->  { x  |  ( x  e.  i  \/  x  e. 
{ i } ) }  e.  { (/) } )
10 elsni 3899 . . . . . 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 2485 . . . 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 2781 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 1364    e. wcel 1761   {cab 2427   A.wral 2713    u. cun 3323   (/)c0 3634   {csn 3874   U_ciun 4168   suc csuc 4717   ` cfv 5415   omcom 6475   1oc1o 6909    predc-bnj14 31510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ral 2718  df-v 2972  df-dif 3328  df-un 3330  df-nul 3635  df-sn 3875  df-suc 4721  df-1o 6916
This theorem is referenced by:  bnj150  31703
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