Theorem frege108d 42278

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 42493. (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 42276	. 2 ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵)
8	7	frege106d 42277	1 ⊢ (𝜑 → (𝐴(t+‘𝑅)𝐵 ∨ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ∨ wo 845   = wceq 1541   ∈ wcel 2106  Vcvv 3473   class class class wbr 5141  ‘cfv 6532  t+ctcl 14914

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-int 4944  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-iota 6484  df-fun 6534  df-fv 6540  df-trcl 14916

This theorem is referenced by:  frege111d  42281