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Theorem dffrege99 40650
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 5084 . 2 (𝑋((t+‘𝑅) ∪ I )𝑍 ↔ (𝑋(t+‘𝑅)𝑍𝑋 I 𝑍))
2 df-or 845 . 2 ((𝑋(t+‘𝑅)𝑍𝑋 I 𝑍) ↔ (¬ 𝑋(t+‘𝑅)𝑍𝑋 I 𝑍))
3 frege99.z . . . . . 6 𝑍𝑈
43elexi 3463 . . . . 5 𝑍 ∈ V
54ideq 5691 . . . 4 (𝑋 I 𝑍𝑋 = 𝑍)
6 eqcom 2808 . . . 4 (𝑋 = 𝑍𝑍 = 𝑋)
75, 6bitri 278 . . 3 (𝑋 I 𝑍𝑍 = 𝑋)
87imbi2i 339 . 2 ((¬ 𝑋(t+‘𝑅)𝑍𝑋 I 𝑍) ↔ (¬ 𝑋(t+‘𝑅)𝑍𝑍 = 𝑋))
91, 2, 83bitrri 301 1 ((¬ 𝑋(t+‘𝑅)𝑍𝑍 = 𝑋) ↔ 𝑋((t+‘𝑅) ∪ I )𝑍)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wo 844   = wceq 1538  wcel 2112  cun 3882   class class class wbr 5033   I cid 5427  cfv 6328  t+ctcl 14340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530
This theorem is referenced by:  frege100  40651  frege105  40656
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