Theorem frege102d 42960

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 43171. (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 42955	. 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 5160	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴𝑅𝐵)
17	13, 16	frege91d 42957	. 2 ⊢ ((𝜑 ∧ 𝐴 = 𝐶) → 𝐴(t+‘𝑅)𝐵)
18		frege102d.ac	. 2 ⊢ (𝜑 → (𝐴(t+‘𝑅)𝐶 ∨ 𝐴 = 𝐶))
19	12, 17, 18	mpjaodan 955	1 ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵)

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 844   = wceq 1533   ∈ wcel 2098  Vcvv 3466   class class class wbr 5138  ‘cfv 6533  t+ctcl 14928

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-iota 6485  df-fun 6535  df-fv 6541  df-trcl 14930

This theorem is referenced by:  frege108d  42962