Theorem frege106 43872

Description: Whatever follows 𝑋 in the 𝑅-sequence belongs to the 𝑅 -sequence beginning with 𝑋. Proposition 106 of [Frege1879] p. 73. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
frege103.z	⊢ 𝑍 ∈ 𝑉

Assertion

Ref	Expression
frege106	⊢ (𝑋(t+‘𝑅)𝑍 → 𝑋((t+‘𝑅) ∪ I )𝑍)

Proof of Theorem frege106

Step	Hyp	Ref	Expression
1		frege103.z	. . 3 ⊢ 𝑍 ∈ 𝑉
2	1	frege105 43871	. 2 ⊢ ((¬ 𝑋(t+‘𝑅)𝑍 → 𝑍 = 𝑋) → 𝑋((t+‘𝑅) ∪ I )𝑍)
3		frege37 43743	. 2 ⊢ (((¬ 𝑋(t+‘𝑅)𝑍 → 𝑍 = 𝑋) → 𝑋((t+‘𝑅) ∪ I )𝑍) → (𝑋(t+‘𝑅)𝑍 → 𝑋((t+‘𝑅) ∪ I )𝑍))
4	2, 3	ax-mp 5	1 ⊢ (𝑋(t+‘𝑅)𝑍 → 𝑋((t+‘𝑅) ∪ I )𝑍)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1537   ∈ wcel 2103   ∪ cun 3968   class class class wbr 5169   I cid 5596  ‘cfv 6572  t+ctcl 15030

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pr 5450  ax-frege1 43693  ax-frege2 43694  ax-frege8 43712  ax-frege28 43733  ax-frege31 43737  ax-frege52a 43760

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-ifp 1064  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-dif 3973  df-un 3975  df-ss 3987  df-nul 4348  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5170  df-opab 5232  df-id 5597  df-xp 5705  df-rel 5706

This theorem is referenced by:  frege107  43873