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Theorem frege91d 40439
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 40642. (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 14363 . . . 4 (𝑅 ∈ V → 𝑅 ⊆ (t+‘𝑅))
42, 3syl 17 . . 3 (𝜑𝑅 ⊆ (t+‘𝑅))
54ssbrd 5076 . 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 2112  Vcvv 3444  wss 3884   class class class wbr 5033  cfv 6328  t+ctcl 14340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-int 4842  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-iota 6287  df-fun 6330  df-fv 6336  df-trcl 14342
This theorem is referenced by:  frege102d  40442  frege129d  40451
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