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Theorem frege111d 43707
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 43922. (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 43704 . 2 (𝜑 → (𝐴(t+‘𝑅)𝐵𝐴 = 𝐵))
87frege114d 43706 1 (𝜑 → (𝐴(t+‘𝑅)𝐵𝐴 = 𝐵𝐵(t+‘𝑅)𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 846  w3o 1084   = wceq 1535  wcel 2104  Vcvv 3477   class class class wbr 5149  cfv 6558  t+ctcl 15010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-10 2137  ax-11 2153  ax-12 2173  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5366  ax-pr 5430  ax-un 7747
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2536  df-eu 2565  df-clab 2711  df-cleq 2725  df-clel 2812  df-nfc 2888  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3433  df-v 3479  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4915  df-int 4954  df-br 5150  df-opab 5212  df-mpt 5233  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 6510  df-fun 6560  df-fv 6566  df-trcl 15012
This theorem is referenced by: (None)
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