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Theorem dffrege99 37173
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 4531 . 2 (𝑋((t+‘𝑅) ∪ I )𝑍 ↔ (𝑋(t+‘𝑅)𝑍𝑋 I 𝑍))
2 df-or 383 . 2 ((𝑋(t+‘𝑅)𝑍𝑋 I 𝑍) ↔ (¬ 𝑋(t+‘𝑅)𝑍𝑋 I 𝑍))
3 frege99.z . . . . . 6 𝑍𝑈
43elexi 3090 . . . . 5 𝑍 ∈ V
54ideq 5088 . . . 4 (𝑋 I 𝑍𝑋 = 𝑍)
6 eqcom 2521 . . . 4 (𝑋 = 𝑍𝑍 = 𝑋)
75, 6bitri 262 . . 3 (𝑋 I 𝑍𝑍 = 𝑋)
87imbi2i 324 . 2 ((¬ 𝑋(t+‘𝑅)𝑍𝑋 I 𝑍) ↔ (¬ 𝑋(t+‘𝑅)𝑍𝑍 = 𝑋))
91, 2, 83bitrri 285 1 ((¬ 𝑋(t+‘𝑅)𝑍𝑍 = 𝑋) ↔ 𝑋((t+‘𝑅) ∪ I )𝑍)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wo 381   = wceq 1474  wcel 1938  cun 3442   class class class wbr 4481   I cid 4842  cfv 5689  t+ctcl 13431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pr 4732
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ral 2805  df-rex 2806  df-rab 2809  df-v 3079  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-sn 4029  df-pr 4031  df-op 4035  df-br 4482  df-opab 4542  df-id 4847  df-xp 4938  df-rel 4939
This theorem is referenced by:  frege100  37174  frege105  37179
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