Theorem frege102d 43743

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 43954. (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 43738	. 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 5169	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴𝑅𝐵)
17	13, 16	frege91d 43740	. 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 1536   ∈ wcel 2105  Vcvv 3477   class class class wbr 5147  ‘cfv 6562  t+ctcl 15020

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-iota 6515  df-fun 6564  df-fv 6570  df-trcl 15022

This theorem is referenced by:  frege108d  43745