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Theorem frege91d 43764
Description: If 𝐵 follows 𝐴 in 𝑅 then 𝐵 follows 𝐴 in the transitive closure of 𝑅. Similar to Proposition 91 of [Frege1879] p. 68. Comparw with frege91 43967. (Contributed by RP, 15-Jul-2020.)
Hypotheses
Ref Expression
frege91d.r (𝜑𝑅 ∈ V)
frege91d.ac (𝜑𝐴𝑅𝐵)
Assertion
Ref Expression
frege91d (𝜑𝐴(t+‘𝑅)𝐵)

Proof of Theorem frege91d
StepHypRef Expression
1 frege91d.ac . 2 (𝜑𝐴𝑅𝐵)
2 frege91d.r . . . 4 (𝜑𝑅 ∈ V)
3 trclfvlb 15047 . . . 4 (𝑅 ∈ V → 𝑅 ⊆ (t+‘𝑅))
42, 3syl 17 . . 3 (𝜑𝑅 ⊆ (t+‘𝑅))
54ssbrd 5186 . 2 (𝜑 → (𝐴𝑅𝐵𝐴(t+‘𝑅)𝐵))
61, 5mpd 15 1 (𝜑𝐴(t+‘𝑅)𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  Vcvv 3480  wss 3951   class class class wbr 5143  cfv 6561  t+ctcl 15024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-iota 6514  df-fun 6563  df-fv 6569  df-trcl 15026
This theorem is referenced by:  frege102d  43767  frege129d  43776
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