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Theorem frege108d 43372
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 43587. (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 43370 . 2 (𝜑𝐴(t+‘𝑅)𝐵)
87frege106d 43371 1 (𝜑 → (𝐴(t+‘𝑅)𝐵𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 845   = wceq 1533  wcel 2098  Vcvv 3461   class class class wbr 5152  cfv 6553  t+ctcl 14985
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5303  ax-nul 5310  ax-pow 5368  ax-pr 5432  ax-un 7745
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4325  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5579  df-xp 5687  df-rel 5688  df-cnv 5689  df-co 5690  df-dm 5691  df-rn 5692  df-res 5693  df-iota 6505  df-fun 6555  df-fv 6561  df-trcl 14987
This theorem is referenced by:  frege111d  43375
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