Theorem frege102d 43716

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 43927. (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 43711	. 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 5188	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴𝑅𝐵)
17	13, 16	frege91d 43713	. 2 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴(t+‘𝑅)𝐵)
18		frege102d.ac	. 2 ⊢ (𝜑 → (𝐴(t+‘𝑅)𝐶 ∨ 𝐴 = 𝐶))
19	12, 17, 18	mpjaodan 959	1 ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵)

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 846   = wceq 1537   ∈ wcel 2108  Vcvv 3488   class class class wbr 5166  ‘cfv 6573  t+ctcl 15034

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-iota 6525  df-fun 6575  df-fv 6581  df-trcl 15036

This theorem is referenced by:  frege108d  43718