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Theorem frege102d 38363
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 𝐵 follows 𝐴 in the transitive closure of 𝑅. Similar to Proposition 102 of [Frege1879] p. 72. Compare with frege102 38576. (Contributed by RP, 15-Jul-2020.)
Hypotheses
Ref Expression
frege102d.r (𝜑𝑅 ∈ V)
frege102d.a (𝜑𝐴 ∈ V)
frege102d.b (𝜑𝐵 ∈ V)
frege102d.c (𝜑𝐶 ∈ V)
frege102d.ac (𝜑 → (𝐴(t+‘𝑅)𝐶𝐴 = 𝐶))
frege102d.cb (𝜑𝐶𝑅𝐵)
Assertion
Ref Expression
frege102d (𝜑𝐴(t+‘𝑅)𝐵)

Proof of Theorem frege102d
StepHypRef Expression
1 frege102d.r . . . 4 (𝜑𝑅 ∈ V)
21adantr 480 . . 3 ((𝜑𝐴(t+‘𝑅)𝐶) → 𝑅 ∈ V)
3 frege102d.a . . . 4 (𝜑𝐴 ∈ V)
43adantr 480 . . 3 ((𝜑𝐴(t+‘𝑅)𝐶) → 𝐴 ∈ V)
5 frege102d.b . . . 4 (𝜑𝐵 ∈ V)
65adantr 480 . . 3 ((𝜑𝐴(t+‘𝑅)𝐶) → 𝐵 ∈ V)
7 frege102d.c . . . 4 (𝜑𝐶 ∈ V)
87adantr 480 . . 3 ((𝜑𝐴(t+‘𝑅)𝐶) → 𝐶 ∈ V)
9 simpr 476 . . 3 ((𝜑𝐴(t+‘𝑅)𝐶) → 𝐴(t+‘𝑅)𝐶)
10 frege102d.cb . . . 4 (𝜑𝐶𝑅𝐵)
1110adantr 480 . . 3 ((𝜑𝐴(t+‘𝑅)𝐶) → 𝐶𝑅𝐵)
122, 4, 6, 8, 9, 11frege96d 38358 . 2 ((𝜑𝐴(t+‘𝑅)𝐶) → 𝐴(t+‘𝑅)𝐵)
131adantr 480 . . 3 ((𝜑𝐴 = 𝐶) → 𝑅 ∈ V)
14 simpr 476 . . . 4 ((𝜑𝐴 = 𝐶) → 𝐴 = 𝐶)
1510adantr 480 . . . 4 ((𝜑𝐴 = 𝐶) → 𝐶𝑅𝐵)
1614, 15eqbrtrd 4707 . . 3 ((𝜑𝐴 = 𝐶) → 𝐴𝑅𝐵)
1713, 16frege91d 38360 . 2 ((𝜑𝐴 = 𝐶) → 𝐴(t+‘𝑅)𝐵)
18 frege102d.ac . 2 (𝜑 → (𝐴(t+‘𝑅)𝐶𝐴 = 𝐶))
1912, 17, 18mpjaodan 844 1 (𝜑𝐴(t+‘𝑅)𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 382  wa 383   = wceq 1523  wcel 2030  Vcvv 3231   class class class wbr 4685  cfv 5926  t+ctcl 13770
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-int 4508  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-iota 5889  df-fun 5928  df-fv 5934  df-trcl 13772
This theorem is referenced by:  frege108d  38365
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