Theorem frege111d 45020

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 𝐴 follows 𝐵 or 𝐵 and 𝐴 in the transitive closure of 𝑅. Similar to Proposition 111 of [Frege1879] p. 75. Compare with frege111 45235. (Contributed by RP, 15-Jul-2020.)

Hypotheses

Ref	Expression
frege111d.r	⊢ (𝜑 → 𝑅 ∈ V)
frege111d.a	⊢ (𝜑 → 𝐴 ∈ V)
frege111d.b	⊢ (𝜑 → 𝐵 ∈ V)
frege111d.c	⊢ (𝜑 → 𝐶 ∈ V)
frege111d.ac	⊢ (𝜑 → (𝐴(t+‘𝑅)𝐶 ∨ 𝐴 = 𝐶))
frege111d.cb	⊢ (𝜑 → 𝐶𝑅𝐵)

Assertion

Ref	Expression
frege111d	⊢ (𝜑 → (𝐴(t+‘𝑅)𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵(t+‘𝑅)𝐴))

Proof of Theorem frege111d

Step	Hyp	Ref	Expression
1		frege111d.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
2		frege111d.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ V)
3		frege111d.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
4		frege111d.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ V)
5		frege111d.ac	. . 3 ⊢ (𝜑 → (𝐴(t+‘𝑅)𝐶 ∨ 𝐴 = 𝐶))
6		frege111d.cb	. . 3 ⊢ (𝜑 → 𝐶𝑅𝐵)
7	1, 2, 3, 4, 5, 6	frege108d 45017	. 2 ⊢ (𝜑 → (𝐴(t+‘𝑅)𝐵 ∨ 𝐴 = 𝐵))
8	7	frege114d 45019	1 ⊢ (𝜑 → (𝐴(t+‘𝑅)𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵(t+‘𝑅)𝐴))

This proof depends on syntax axioms:   → wi 4   ∨ wo 861   ∨ w3o 1102   = wceq 1570   ∈ wcel 2147  Vcvv 3454   class class class wbr 5108  ‘cfv 6566  t+ctcl 15181

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-iota 6520  df-fun 6568  df-fv 6574  df-trcl 15183

This theorem is used by: (None)