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Theorem frege114 37291
Description: If 𝑋 belongs to the 𝑅-sequence beginning with 𝑍, then 𝑍 belongs to the 𝑅-sequence beginning with 𝑋 or 𝑋 follows 𝑍 in the 𝑅-sequence. Proposition 114 of [Frege1879] p. 76. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
frege114.x 𝑋𝑈
frege114.z 𝑍𝑉
Assertion
Ref Expression
frege114 (𝑍((t+‘𝑅) ∪ I )𝑋 → (¬ 𝑍(t+‘𝑅)𝑋𝑋((t+‘𝑅) ∪ I )𝑍))

Proof of Theorem frege114
StepHypRef Expression
1 frege114.x . . 3 𝑋𝑈
21frege104 37281 . 2 (𝑍((t+‘𝑅) ∪ I )𝑋 → (¬ 𝑍(t+‘𝑅)𝑋𝑍 = 𝑋))
3 frege114.z . . 3 𝑍𝑉
43frege113 37290 . 2 ((𝑍((t+‘𝑅) ∪ I )𝑋 → (¬ 𝑍(t+‘𝑅)𝑋𝑍 = 𝑋)) → (𝑍((t+‘𝑅) ∪ I )𝑋 → (¬ 𝑍(t+‘𝑅)𝑋𝑋((t+‘𝑅) ∪ I )𝑍)))
52, 4ax-mp 5 1 (𝑍((t+‘𝑅) ∪ I )𝑋 → (¬ 𝑍(t+‘𝑅)𝑋𝑋((t+‘𝑅) ∪ I )𝑍))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1475  wcel 1977  cun 3538   class class class wbr 4583   I cid 4948  cfv 5804  t+ctcl 13572
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-frege1 37104  ax-frege2 37105  ax-frege8 37123  ax-frege52a 37171  ax-frege52c 37202
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ifp 1007  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045
This theorem is referenced by:  frege126  37303
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