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	⊢ 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 41535	. 2 ⊢ (𝑍((t+‘𝑅) ∪ I )𝑋 → (¬ 𝑍(t+‘𝑅)𝑋 → 𝑍 = 𝑋))
3		frege114.z	. . 3 ⊢ 𝑍 ∈ 𝑉
4	3	frege113 41544	. 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))

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