Theorem frege108d 43704

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 43919. (Contributed by RP, 15-Jul-2020.)

Hypotheses

Ref	Expression
frege108d.r	⊢ (𝜑 → 𝑅 ∈ V)
frege108d.a	⊢ (𝜑 → 𝐴 ∈ V)
frege108d.b	⊢ (𝜑 → 𝐵 ∈ V)
frege108d.c	⊢ (𝜑 → 𝐶 ∈ V)
frege108d.ac	⊢ (𝜑 → (𝐴(t+‘𝑅)𝐶 ∨ 𝐴 = 𝐶))
frege108d.cb	⊢ (𝜑 → 𝐶𝑅𝐵)

Assertion

Ref	Expression
frege108d	⊢ (𝜑 → (𝐴(t+‘𝑅)𝐵 ∨ 𝐴 = 𝐵))

Proof of Theorem frege108d

Step	Hyp	Ref	Expression
1		frege108d.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
2		frege108d.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ V)
3		frege108d.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
4		frege108d.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ V)
5		frege108d.ac	. . 3 ⊢ (𝜑 → (𝐴(t+‘𝑅)𝐶 ∨ 𝐴 = 𝐶))
6		frege108d.cb	. . 3 ⊢ (𝜑 → 𝐶𝑅𝐵)
7	1, 2, 3, 4, 5, 6	frege102d 43702	. 2 ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵)
8	7	frege106d 43703	1 ⊢ (𝜑 → (𝐴(t+‘𝑅)𝐵 ∨ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ∨ wo 846   = wceq 1535   ∈ wcel 2104  Vcvv 3477   class class class wbr 5149  ‘cfv 6558  t+ctcl 15010

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-10 2137  ax-11 2153  ax-12 2173  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5366  ax-pr 5430  ax-un 7747

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2536  df-eu 2565  df-clab 2711  df-cleq 2725  df-clel 2812  df-nfc 2888  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3433  df-v 3479  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4915  df-int 4954  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-iota 6510  df-fun 6560  df-fv 6566  df-trcl 15012

This theorem is referenced by:  frege111d  43707