Theorem frege114 45120

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	⊢ 𝑋 ∈ 𝑈
frege114.z	⊢ 𝑍 ∈ 𝑉

Assertion

Ref	Expression
frege114	⊢ (𝑍((t+‘𝑅) ∪ I )𝑋 → (¬ 𝑍(t+‘𝑅)𝑋 → 𝑋((t+‘𝑅) ∪ I )𝑍))

Proof of Theorem frege114

Step	Hyp	Ref	Expression
1		frege114.x	. . 3 ⊢ 𝑋 ∈ 𝑈
2	1	frege104 45110	. 2 ⊢ (𝑍((t+‘𝑅) ∪ I )𝑋 → (¬ 𝑍(t+‘𝑅)𝑋 → 𝑍 = 𝑋))
3		frege114.z	. . 3 ⊢ 𝑍 ∈ 𝑉
4	3	frege113 45119	. 2 ⊢ ((𝑍((t+‘𝑅) ∪ I )𝑋 → (¬ 𝑍(t+‘𝑅)𝑋 → 𝑍 = 𝑋)) → (𝑍((t+‘𝑅) ∪ I )𝑋 → (¬ 𝑍(t+‘𝑅)𝑋 → 𝑋((t+‘𝑅) ∪ I )𝑍)))
5	2, 4	ax-mp 5	1 ⊢ (𝑍((t+‘𝑅) ∪ I )𝑋 → (¬ 𝑍(t+‘𝑅)𝑋 → 𝑋((t+‘𝑅) ∪ I )𝑍))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1569   ∈ wcel 2146   ∪ cun 4152   class class class wbr 5106   I cid 5549  ‘cfv 6547  t+ctcl 15153

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-12 2215  ax-ext 2735  ax-sep 5252  ax-pr 5395  ax-frege1 44933  ax-frege2 44934  ax-frege8 44952  ax-frege52a 45000  ax-frege52c 45031

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ifp 1079  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662

This theorem is used by:  frege126  45132