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Theorem frege108d 41345
Description: If either 𝐴 and 𝐶 are the same or 𝐶 follows 𝐴 in the transitive closure of 𝑅 and 𝐵 is the successor to 𝐶, then either 𝐴 and 𝐵 are the same or 𝐵 follows 𝐴 in the transitive closure of 𝑅. Similar to Proposition 108 of [Frege1879] p. 74. Compare with frege108 41560. (Contributed by RP, 15-Jul-2020.)
Hypotheses
Ref Expression
frege108d.r (𝜑𝑅 ∈ V)
frege108d.a (𝜑𝐴 ∈ V)
frege108d.b (𝜑𝐵 ∈ V)
frege108d.c (𝜑𝐶 ∈ V)
frege108d.ac (𝜑 → (𝐴(t+‘𝑅)𝐶𝐴 = 𝐶))
frege108d.cb (𝜑𝐶𝑅𝐵)
Assertion
Ref Expression
frege108d (𝜑 → (𝐴(t+‘𝑅)𝐵𝐴 = 𝐵))

Proof of Theorem frege108d
StepHypRef Expression
1 frege108d.r . . 3 (𝜑𝑅 ∈ V)
2 frege108d.a . . 3 (𝜑𝐴 ∈ V)
3 frege108d.b . . 3 (𝜑𝐵 ∈ V)
4 frege108d.c . . 3 (𝜑𝐶 ∈ V)
5 frege108d.ac . . 3 (𝜑 → (𝐴(t+‘𝑅)𝐶𝐴 = 𝐶))
6 frege108d.cb . . 3 (𝜑𝐶𝑅𝐵)
71, 2, 3, 4, 5, 6frege102d 41343 . 2 (𝜑𝐴(t+‘𝑅)𝐵)
87frege106d 41344 1 (𝜑 → (𝐴(t+‘𝑅)𝐵𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 844   = wceq 1539  wcel 2106  Vcvv 3429   class class class wbr 5073  cfv 6426  t+ctcl 14706
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5221  ax-nul 5228  ax-pow 5286  ax-pr 5350  ax-un 7578
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3431  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-br 5074  df-opab 5136  df-mpt 5157  df-id 5484  df-xp 5590  df-rel 5591  df-cnv 5592  df-co 5593  df-dm 5594  df-rn 5595  df-res 5596  df-iota 6384  df-fun 6428  df-fv 6434  df-trcl 14708
This theorem is referenced by:  frege111d  41348
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