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Theorem frege111d 36869
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 37087. (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 36866 . 2 (𝜑 → (𝐴(t+‘𝑅)𝐵𝐴 = 𝐵))
87frege114d 36868 1 (𝜑 → (𝐴(t+‘𝑅)𝐵𝐴 = 𝐵𝐵(t+‘𝑅)𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 381  w3o 1029   = wceq 1474  wcel 1975  Vcvv 3168   class class class wbr 4573  cfv 5786  t+ctcl 13514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-sbc 3398  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-int 4401  df-br 4574  df-opab 4634  df-mpt 4635  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-iota 5750  df-fun 5788  df-fv 5794  df-trcl 13516
This theorem is referenced by: (None)
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