Theorem frege102d 41343

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 41554. (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 481	. . 3 ⊢ ((𝜑 ∧ 𝐴(t+‘𝑅)𝐶) → 𝑅 ∈ V)
3		frege102d.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ V)
4	3	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝐴(t+‘𝑅)𝐶) → 𝐴 ∈ V)
5		frege102d.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ V)
6	5	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝐴(t+‘𝑅)𝐶) → 𝐵 ∈ V)
7		frege102d.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ V)
8	7	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝐴(t+‘𝑅)𝐶) → 𝐶 ∈ V)
9		simpr 485	. . 3 ⊢ ((𝜑 ∧ 𝐴(t+‘𝑅)𝐶) → 𝐴(t+‘𝑅)𝐶)
10		frege102d.cb	. . . 4 ⊢ (𝜑 → 𝐶𝑅𝐵)
11	10	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝐴(t+‘𝑅)𝐶) → 𝐶𝑅𝐵)
12	2, 4, 6, 8, 9, 11	frege96d 41338	. 2 ⊢ ((𝜑 ∧ 𝐴(t+‘𝑅)𝐶) → 𝐴(t+‘𝑅)𝐵)
13	1	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝑅 ∈ V)
14		simpr 485	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴 = 𝐶)
15	10	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐶𝑅𝐵)
16	14, 15	eqbrtrd 5095	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴𝑅𝐵)
17	13, 16	frege91d 41340	. 2 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴(t+‘𝑅)𝐵)
18		frege102d.ac	. 2 ⊢ (𝜑 → (𝐴(t+‘𝑅)𝐶 ∨ 𝐴 = 𝐶))
19	12, 17, 18	mpjaodan 956	1 ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵)

Syntax hints:   → wi 4   ∧ wa 396   ∨ wo 844   = wceq 1539   ∈ wcel 2106  Vcvv 3429   class class class wbr 5073  ‘cfv 6426  t+ctcl 14706

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5221  ax-nul 5228  ax-pow 5286  ax-pr 5350  ax-un 7578

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3431  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-br 5074  df-opab 5136  df-mpt 5157  df-id 5484  df-xp 5590  df-rel 5591  df-cnv 5592  df-co 5593  df-dm 5594  df-rn 5595  df-res 5596  df-iota 6384  df-fun 6428  df-fv 6434  df-trcl 14708

This theorem is referenced by:  frege108d  41345