Theorem frege108d 44332

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 44547. (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 44330	. 2 ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵)
8	7	frege106d 44331	1 ⊢ (𝜑 → (𝐴(t+‘𝑅)𝐵 ∨ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ∨ wo 858   = wceq 1560   ∈ wcel 2142  Vcvv 3454   class class class wbr 5100  ‘cfv 6521  t+ctcl 14998

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4906  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-iota 6477  df-fun 6523  df-fv 6529  df-trcl 15000

This theorem is referenced by:  frege111d  44335