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Theorem frege106 39103
 Description: Whatever follows 𝑋 in the 𝑅-sequence belongs to the 𝑅 -sequence beginning with 𝑋. Proposition 106 of [Frege1879] p. 73. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
frege103.z 𝑍𝑉
Assertion
Ref Expression
frege106 (𝑋(t+‘𝑅)𝑍𝑋((t+‘𝑅) ∪ I )𝑍)

Proof of Theorem frege106
StepHypRef Expression
1 frege103.z . . 3 𝑍𝑉
21frege105 39102 . 2 ((¬ 𝑋(t+‘𝑅)𝑍𝑍 = 𝑋) → 𝑋((t+‘𝑅) ∪ I )𝑍)
3 frege37 38974 . 2 (((¬ 𝑋(t+‘𝑅)𝑍𝑍 = 𝑋) → 𝑋((t+‘𝑅) ∪ I )𝑍) → (𝑋(t+‘𝑅)𝑍𝑋((t+‘𝑅) ∪ I )𝑍))
42, 3ax-mp 5 1 (𝑋(t+‘𝑅)𝑍𝑋((t+‘𝑅) ∪ I )𝑍)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1658   ∈ wcel 2166   ∪ cun 3796   class class class wbr 4873   I cid 5249  ‘cfv 6123  t+ctcl 14103 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127  ax-frege1 38924  ax-frege2 38925  ax-frege8 38943  ax-frege28 38964  ax-frege31 38968  ax-frege52a 38991 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-ifp 1092  df-3an 1115  df-tru 1662  df-fal 1672  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-br 4874  df-opab 4936  df-id 5250  df-xp 5348  df-rel 5349 This theorem is referenced by:  frege107  39104
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