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Theorem frege98d 40828
 Description: If 𝐶 follows 𝐴 and 𝐵 follows 𝐶 in the transitive closure of 𝑅, then 𝐵 follows 𝐴 in the transitive closure of 𝑅. Similar to Proposition 98 of [Frege1879] p. 71. Compare with frege98 41036. (Contributed by RP, 15-Jul-2020.)
Hypotheses
Ref Expression
frege98d.a (𝜑𝐴 ∈ V)
frege98d.b (𝜑𝐵 ∈ V)
frege98d.c (𝜑𝐶 ∈ V)
frege98d.ac (𝜑𝐴(t+‘𝑅)𝐶)
frege98d.cb (𝜑𝐶(t+‘𝑅)𝐵)
Assertion
Ref Expression
frege98d (𝜑𝐴(t+‘𝑅)𝐵)

Proof of Theorem frege98d
StepHypRef Expression
1 frege98d.a . . 3 (𝜑𝐴 ∈ V)
2 frege98d.b . . 3 (𝜑𝐵 ∈ V)
3 frege98d.c . . 3 (𝜑𝐶 ∈ V)
4 frege98d.ac . . 3 (𝜑𝐴(t+‘𝑅)𝐶)
5 frege98d.cb . . 3 (𝜑𝐶(t+‘𝑅)𝐵)
6 brcogw 5709 . . 3 (((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) ∧ (𝐴(t+‘𝑅)𝐶𝐶(t+‘𝑅)𝐵)) → 𝐴((t+‘𝑅) ∘ (t+‘𝑅))𝐵)
71, 2, 3, 4, 5, 6syl32anc 1376 . 2 (𝜑𝐴((t+‘𝑅) ∘ (t+‘𝑅))𝐵)
8 trclfvcotrg 14424 . . . 4 ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)
98a1i 11 . . 3 (𝜑 → ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))
109ssbrd 5076 . 2 (𝜑 → (𝐴((t+‘𝑅) ∘ (t+‘𝑅))𝐵𝐴(t+‘𝑅)𝐵))
117, 10mpd 15 1 (𝜑𝐴(t+‘𝑅)𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2112  Vcvv 3410   ⊆ wss 3859   class class class wbr 5033   ∘ ccom 5529  ‘cfv 6336  t+ctcl 14393 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3698  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-int 4840  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-iota 6295  df-fun 6338  df-fv 6344  df-trcl 14395 This theorem is referenced by: (None)
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