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Theorem frege111d 42281
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 𝐴 follows 𝐵 or 𝐵 and 𝐴 in the transitive closure of 𝑅. Similar to Proposition 111 of [Frege1879] p. 75. Compare with frege111 42496. (Contributed by RP, 15-Jul-2020.)
Hypotheses
Ref Expression
frege111d.r (𝜑𝑅 ∈ V)
frege111d.a (𝜑𝐴 ∈ V)
frege111d.b (𝜑𝐵 ∈ V)
frege111d.c (𝜑𝐶 ∈ V)
frege111d.ac (𝜑 → (𝐴(t+‘𝑅)𝐶𝐴 = 𝐶))
frege111d.cb (𝜑𝐶𝑅𝐵)
Assertion
Ref Expression
frege111d (𝜑 → (𝐴(t+‘𝑅)𝐵𝐴 = 𝐵𝐵(t+‘𝑅)𝐴))

Proof of Theorem frege111d
StepHypRef Expression
1 frege111d.r . . 3 (𝜑𝑅 ∈ V)
2 frege111d.a . . 3 (𝜑𝐴 ∈ V)
3 frege111d.b . . 3 (𝜑𝐵 ∈ V)
4 frege111d.c . . 3 (𝜑𝐶 ∈ V)
5 frege111d.ac . . 3 (𝜑 → (𝐴(t+‘𝑅)𝐶𝐴 = 𝐶))
6 frege111d.cb . . 3 (𝜑𝐶𝑅𝐵)
71, 2, 3, 4, 5, 6frege108d 42278 . 2 (𝜑 → (𝐴(t+‘𝑅)𝐵𝐴 = 𝐵))
87frege114d 42280 1 (𝜑 → (𝐴(t+‘𝑅)𝐵𝐴 = 𝐵𝐵(t+‘𝑅)𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 845  w3o 1086   = wceq 1541  wcel 2106  Vcvv 3473   class class class wbr 5141  cfv 6532  t+ctcl 14914
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-int 4944  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-iota 6484  df-fun 6534  df-fv 6540  df-trcl 14916
This theorem is referenced by: (None)
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