Theorem frege114 43935

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 43925	. 2 ⊢ (𝑍((t+‘𝑅) ∪ I )𝑋 → (¬ 𝑍(t+‘𝑅)𝑋 → 𝑍 = 𝑋))
3		frege114.z	. . 3 ⊢ 𝑍 ∈ 𝑉
4	3	frege113 43934	. 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 2107   ∪ cun 3931   class class class wbr 5125   I cid 5559  ‘cfv 6542  t+ctcl 15007

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-12 2176  ax-ext 2706  ax-sep 5278  ax-nul 5288  ax-pr 5414  ax-frege1 43748  ax-frege2 43749  ax-frege8 43767  ax-frege52a 43815  ax-frege52c 43846

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ral 3051  df-rex 3060  df-rab 3421  df-v 3466  df-sbc 3773  df-dif 3936  df-un 3938  df-ss 3950  df-nul 4316  df-if 4508  df-sn 4609  df-pr 4611  df-op 4615  df-br 5126  df-opab 5188  df-id 5560  df-xp 5673  df-rel 5674

This theorem is referenced by:  frege126  43947