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Theorem dffrege99 36528
Description: If  Z is identical with  X or follows  X in the  R -sequence, then we say : " Z belongs to the 
R-sequence beginning with  X " or " X belongs to the  R-sequence ending with  Z". Definition 99 of [Frege1879] p. 71. (Contributed by RP, 2-Jul-2020.)
Hypothesis
Ref Expression
frege99.z  |-  Z  e.  U
Assertion
Ref Expression
dffrege99  |-  ( ( -.  X ( t+ `  R ) Z  ->  Z  =  X )  <->  X (
( t+ `  R )  u.  _I  ) Z )

Proof of Theorem dffrege99
StepHypRef Expression
1 brun 4472 . 2  |-  ( X ( ( t+ `  R )  u.  _I  ) Z  <->  ( X
( t+ `  R ) Z  \/  X  _I  Z )
)
2 df-or 371 . 2  |-  ( ( X ( t+ `  R ) Z  \/  X  _I  Z
)  <->  ( -.  X
( t+ `  R ) Z  ->  X  _I  Z )
)
3 frege99.z . . . . . 6  |-  Z  e.  U
43elexi 3090 . . . . 5  |-  Z  e. 
_V
54ideq 5006 . . . 4  |-  ( X  _I  Z  <->  X  =  Z )
6 eqcom 2431 . . . 4  |-  ( X  =  Z  <->  Z  =  X )
75, 6bitri 252 . . 3  |-  ( X  _I  Z  <->  Z  =  X )
87imbi2i 313 . 2  |-  ( ( -.  X ( t+ `  R ) Z  ->  X  _I  Z )  <->  ( -.  X ( t+ `  R ) Z  ->  Z  =  X ) )
91, 2, 83bitrri 275 1  |-  ( ( -.  X ( t+ `  R ) Z  ->  Z  =  X )  <->  X (
( t+ `  R )  u.  _I  ) Z )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    = wceq 1437    e. wcel 1872    u. cun 3434   class class class wbr 4423    _I cid 4763   ` cfv 5601   t+ctcl 13049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-br 4424  df-opab 4483  df-id 4768  df-xp 4859  df-rel 4860
This theorem is referenced by:  frege100  36529  frege105  36534
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