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Theorem frege114 41545
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 41535 . 2 (𝑍((t+‘𝑅) ∪ I )𝑋 → (¬ 𝑍(t+‘𝑅)𝑋𝑍 = 𝑋))
3 frege114.z . . 3 𝑍𝑉
43frege113 41544 . 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 1539  wcel 2106  cun 3886   class class class wbr 5075   I cid 5485  cfv 6428  t+ctcl 14685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-frege1 41358  ax-frege2 41359  ax-frege8 41377  ax-frege52a 41425  ax-frege52c 41456
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-ifp 1061  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3433  df-sbc 3718  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4259  df-if 4462  df-sn 4564  df-pr 4566  df-op 4570  df-br 5076  df-opab 5138  df-id 5486  df-xp 5592  df-rel 5593
This theorem is referenced by:  frege126  41557
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