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Theorem frege111d 43750
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 43965. (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 43747 . 2 (𝜑 → (𝐴(t+‘𝑅)𝐵𝐴 = 𝐵))
87frege114d 43749 1 (𝜑 → (𝐴(t+‘𝑅)𝐵𝐴 = 𝐵𝐵(t+‘𝑅)𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 848  w3o 1086   = wceq 1540  wcel 2108  Vcvv 3479   class class class wbr 5141  cfv 6559  t+ctcl 15020
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5294  ax-nul 5304  ax-pow 5363  ax-pr 5430  ax-un 7751
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-int 4945  df-br 5142  df-opab 5204  df-mpt 5224  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-iota 6512  df-fun 6561  df-fv 6567  df-trcl 15022
This theorem is referenced by: (None)
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