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Theorem frege108d 38888
 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 39104. (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 38886 . 2 (𝜑𝐴(t+‘𝑅)𝐵)
87frege106d 38887 1 (𝜑 → (𝐴(t+‘𝑅)𝐵𝐴 = 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 878   = wceq 1656   ∈ wcel 2164  Vcvv 3414   class class class wbr 4875  ‘cfv 6127  t+ctcl 14110 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-int 4700  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-iota 6090  df-fun 6129  df-fv 6135  df-trcl 14112 This theorem is referenced by:  frege111d  38891
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