Theorem frege108d 43939

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 44154. (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 43937	. 2 ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵)
8	7	frege106d 43938	1 ⊢ (𝜑 → (𝐴(t+‘𝑅)𝐵 ∨ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ∨ wo 847   = wceq 1541   ∈ wcel 2113  Vcvv 3438   class class class wbr 5096  ‘cfv 6490  t+ctcl 14906

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-iota 6446  df-fun 6492  df-fv 6498  df-trcl 14908

This theorem is referenced by:  frege111d  43942