Theorem frege114 43286

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 43276	. 2 ⊢ (𝑍((t+‘𝑅) ∪ I )𝑋 → (¬ 𝑍(t+‘𝑅)𝑋 → 𝑍 = 𝑋))
3		frege114.z	. . 3 ⊢ 𝑍 ∈ 𝑉
4	3	frege113 43285	. 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 1533   ∈ wcel 2098   ∪ cun 3941   class class class wbr 5141   I cid 5566  ‘cfv 6536  t+ctcl 14935

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-frege1 43099  ax-frege2 43100  ax-frege8 43118  ax-frege52a 43166  ax-frege52c 43197

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ifp 1060  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676

This theorem is referenced by:  frege126  43298