Theorem frege102d 43857

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 𝐵 follows 𝐴 in the transitive closure of 𝑅. Similar to Proposition 102 of [Frege1879] p. 72. Compare with frege102 44068. (Contributed by RP, 15-Jul-2020.)

Hypotheses

Ref	Expression
frege102d.r	⊢ (𝜑 → 𝑅 ∈ V)
frege102d.a	⊢ (𝜑 → 𝐴 ∈ V)
frege102d.b	⊢ (𝜑 → 𝐵 ∈ V)
frege102d.c	⊢ (𝜑 → 𝐶 ∈ V)
frege102d.ac	⊢ (𝜑 → (𝐴(t+‘𝑅)𝐶 ∨ 𝐴 = 𝐶))
frege102d.cb	⊢ (𝜑 → 𝐶𝑅𝐵)

Assertion

Ref	Expression
frege102d	⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵)

Proof of Theorem frege102d

Step	Hyp	Ref	Expression
1		frege102d.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ V)
2	1	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐴(t+‘𝑅)𝐶) → 𝑅 ∈ V)
3		frege102d.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ V)
4	3	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐴(t+‘𝑅)𝐶) → 𝐴 ∈ V)
5		frege102d.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ V)
6	5	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐴(t+‘𝑅)𝐶) → 𝐵 ∈ V)
7		frege102d.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ V)
8	7	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐴(t+‘𝑅)𝐶) → 𝐶 ∈ V)
9		simpr 484	. . 3 ⊢ ((𝜑 ∧ 𝐴(t+‘𝑅)𝐶) → 𝐴(t+‘𝑅)𝐶)
10		frege102d.cb	. . . 4 ⊢ (𝜑 → 𝐶𝑅𝐵)
11	10	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐴(t+‘𝑅)𝐶) → 𝐶𝑅𝐵)
12	2, 4, 6, 8, 9, 11	frege96d 43852	. 2 ⊢ ((𝜑 ∧ 𝐴(t+‘𝑅)𝐶) → 𝐴(t+‘𝑅)𝐵)
13	1	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝑅 ∈ V)
14		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴 = 𝐶)
15	10	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐶𝑅𝐵)
16	14, 15	eqbrtrd 5111	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴𝑅𝐵)
17	13, 16	frege91d 43854	. 2 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴(t+‘𝑅)𝐵)
18		frege102d.ac	. 2 ⊢ (𝜑 → (𝐴(t+‘𝑅)𝐶 ∨ 𝐴 = 𝐶))
19	12, 17, 18	mpjaodan 960	1 ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵)

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2111  Vcvv 3436   class class class wbr 5089  ‘cfv 6481  t+ctcl 14892

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-iota 6437  df-fun 6483  df-fv 6489  df-trcl 14894

This theorem is referenced by:  frege108d  43859