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Theorem dffrege99 43865
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 𝑍𝑈
Assertion
Ref Expression
dffrege99 ((¬ 𝑋(t+‘𝑅)𝑍𝑍 = 𝑋) ↔ 𝑋((t+‘𝑅) ∪ I )𝑍)

Proof of Theorem dffrege99
StepHypRef Expression
1 brun 5220 . 2 (𝑋((t+‘𝑅) ∪ I )𝑍 ↔ (𝑋(t+‘𝑅)𝑍𝑋 I 𝑍))
2 df-or 847 . 2 ((𝑋(t+‘𝑅)𝑍𝑋 I 𝑍) ↔ (¬ 𝑋(t+‘𝑅)𝑍𝑋 I 𝑍))
3 frege99.z . . . . . 6 𝑍𝑈
43elexi 3506 . . . . 5 𝑍 ∈ V
54ideq 5876 . . . 4 (𝑋 I 𝑍𝑋 = 𝑍)
6 eqcom 2741 . . . 4 (𝑋 = 𝑍𝑍 = 𝑋)
75, 6bitri 275 . . 3 (𝑋 I 𝑍𝑍 = 𝑋)
87imbi2i 336 . 2 ((¬ 𝑋(t+‘𝑅)𝑍𝑋 I 𝑍) ↔ (¬ 𝑋(t+‘𝑅)𝑍𝑍 = 𝑋))
91, 2, 83bitrri 298 1 ((¬ 𝑋(t+‘𝑅)𝑍𝑍 = 𝑋) ↔ 𝑋((t+‘𝑅) ∪ I )𝑍)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wo 846   = wceq 1537  wcel 2103  cun 3968   class class class wbr 5169   I cid 5596  cfv 6572  t+ctcl 15030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pr 5450
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-dif 3973  df-un 3975  df-ss 3987  df-nul 4348  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5170  df-opab 5232  df-id 5597  df-xp 5705  df-rel 5706
This theorem is referenced by:  frege100  43866  frege105  43871
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