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Theorem frege91d 36861
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 37067. (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 13539 . . . 4 (𝑅 ∈ V → 𝑅 ⊆ (t+‘𝑅))
42, 3syl 17 . . 3 (𝜑𝑅 ⊆ (t+‘𝑅))
54ssbrd 4616 . 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 1975  Vcvv 3168  wss 3535   class class class wbr 4573  cfv 5786  t+ctcl 13514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-sbc 3398  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-int 4401  df-br 4574  df-opab 4634  df-mpt 4635  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-iota 5750  df-fun 5788  df-fv 5794  df-trcl 13516
This theorem is referenced by:  frege102d  36864  frege129d  36873
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