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Theorem frege102d 40386
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 40598. (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 484 . . 3 ((𝜑𝐴(t+‘𝑅)𝐶) → 𝑅 ∈ V)
3 frege102d.a . . . 4 (𝜑𝐴 ∈ V)
43adantr 484 . . 3 ((𝜑𝐴(t+‘𝑅)𝐶) → 𝐴 ∈ V)
5 frege102d.b . . . 4 (𝜑𝐵 ∈ V)
65adantr 484 . . 3 ((𝜑𝐴(t+‘𝑅)𝐶) → 𝐵 ∈ V)
7 frege102d.c . . . 4 (𝜑𝐶 ∈ V)
87adantr 484 . . 3 ((𝜑𝐴(t+‘𝑅)𝐶) → 𝐶 ∈ V)
9 simpr 488 . . 3 ((𝜑𝐴(t+‘𝑅)𝐶) → 𝐴(t+‘𝑅)𝐶)
10 frege102d.cb . . . 4 (𝜑𝐶𝑅𝐵)
1110adantr 484 . . 3 ((𝜑𝐴(t+‘𝑅)𝐶) → 𝐶𝑅𝐵)
122, 4, 6, 8, 9, 11frege96d 40381 . 2 ((𝜑𝐴(t+‘𝑅)𝐶) → 𝐴(t+‘𝑅)𝐵)
131adantr 484 . . 3 ((𝜑𝐴 = 𝐶) → 𝑅 ∈ V)
14 simpr 488 . . . 4 ((𝜑𝐴 = 𝐶) → 𝐴 = 𝐶)
1510adantr 484 . . . 4 ((𝜑𝐴 = 𝐶) → 𝐶𝑅𝐵)
1614, 15eqbrtrd 5064 . . 3 ((𝜑𝐴 = 𝐶) → 𝐴𝑅𝐵)
1713, 16frege91d 40383 . 2 ((𝜑𝐴 = 𝐶) → 𝐴(t+‘𝑅)𝐵)
18 frege102d.ac . 2 (𝜑 → (𝐴(t+‘𝑅)𝐶𝐴 = 𝐶))
1912, 17, 18mpjaodan 956 1 (𝜑𝐴(t+‘𝑅)𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wo 844   = wceq 1538  wcel 2114  Vcvv 3469   class class class wbr 5042  cfv 6334  t+ctcl 14336
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-int 4852  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-iota 6293  df-fun 6336  df-fv 6342  df-trcl 14338
This theorem is referenced by:  frege108d  40388
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