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Theorem frmin 9744
Description: Every (possibly proper) subclass of a class 𝐴 with a well-founded set-like relation 𝑅 has a minimal element. This is a very strong generalization of tz6.26 6349 and tz7.5 6386. (Contributed by Scott Fenton, 4-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) (Revised by Scott Fenton, 27-Nov-2024.)
Assertion
Ref Expression
frmin (((𝑅 Fr 𝐴𝑅 Se 𝐴) ∧ (𝐵𝐴𝐵 ≠ ∅)) → ∃𝑦𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅)
Distinct variable groups:   𝑦,𝐵   𝑦,𝑅
Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem frmin
Dummy variables 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frss 5644 . . . 4 (𝐵𝐴 → (𝑅 Fr 𝐴𝑅 Fr 𝐵))
2 sess2 5646 . . . 4 (𝐵𝐴 → (𝑅 Se 𝐴𝑅 Se 𝐵))
31, 2anim12d 610 . . 3 (𝐵𝐴 → ((𝑅 Fr 𝐴𝑅 Se 𝐴) → (𝑅 Fr 𝐵𝑅 Se 𝐵)))
4 n0 4347 . . . 4 (𝐵 ≠ ∅ ↔ ∃𝑏 𝑏𝐵)
5 predeq3 6305 . . . . . . . . . . 11 (𝑦 = 𝑏 → Pred(𝑅, 𝐵, 𝑦) = Pred(𝑅, 𝐵, 𝑏))
65eqeq1d 2735 . . . . . . . . . 10 (𝑦 = 𝑏 → (Pred(𝑅, 𝐵, 𝑦) = ∅ ↔ Pred(𝑅, 𝐵, 𝑏) = ∅))
76rspcev 3613 . . . . . . . . 9 ((𝑏𝐵 ∧ Pred(𝑅, 𝐵, 𝑏) = ∅) → ∃𝑦𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅)
87ex 414 . . . . . . . 8 (𝑏𝐵 → (Pred(𝑅, 𝐵, 𝑏) = ∅ → ∃𝑦𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅))
98adantl 483 . . . . . . 7 (((𝑅 Fr 𝐵𝑅 Se 𝐵) ∧ 𝑏𝐵) → (Pred(𝑅, 𝐵, 𝑏) = ∅ → ∃𝑦𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅))
10 predres 6341 . . . . . . . . . . 11 Pred(𝑅, 𝐵, 𝑏) = Pred((𝑅𝐵), 𝐵, 𝑏)
11 relres 6011 . . . . . . . . . . . . 13 Rel (𝑅𝐵)
12 ssttrcl 9710 . . . . . . . . . . . . 13 (Rel (𝑅𝐵) → (𝑅𝐵) ⊆ t++(𝑅𝐵))
1311, 12ax-mp 5 . . . . . . . . . . . 12 (𝑅𝐵) ⊆ t++(𝑅𝐵)
14 predrelss 6339 . . . . . . . . . . . 12 ((𝑅𝐵) ⊆ t++(𝑅𝐵) → Pred((𝑅𝐵), 𝐵, 𝑏) ⊆ Pred(t++(𝑅𝐵), 𝐵, 𝑏))
1513, 14ax-mp 5 . . . . . . . . . . 11 Pred((𝑅𝐵), 𝐵, 𝑏) ⊆ Pred(t++(𝑅𝐵), 𝐵, 𝑏)
1610, 15eqsstri 4017 . . . . . . . . . 10 Pred(𝑅, 𝐵, 𝑏) ⊆ Pred(t++(𝑅𝐵), 𝐵, 𝑏)
17 ssn0 4401 . . . . . . . . . 10 ((Pred(𝑅, 𝐵, 𝑏) ⊆ Pred(t++(𝑅𝐵), 𝐵, 𝑏) ∧ Pred(𝑅, 𝐵, 𝑏) ≠ ∅) → Pred(t++(𝑅𝐵), 𝐵, 𝑏) ≠ ∅)
1816, 17mpan 689 . . . . . . . . 9 (Pred(𝑅, 𝐵, 𝑏) ≠ ∅ → Pred(t++(𝑅𝐵), 𝐵, 𝑏) ≠ ∅)
19 predss 6309 . . . . . . . . 9 Pred(t++(𝑅𝐵), 𝐵, 𝑏) ⊆ 𝐵
2018, 19jctil 521 . . . . . . . 8 (Pred(𝑅, 𝐵, 𝑏) ≠ ∅ → (Pred(t++(𝑅𝐵), 𝐵, 𝑏) ⊆ 𝐵 ∧ Pred(t++(𝑅𝐵), 𝐵, 𝑏) ≠ ∅))
21 dffr4 6321 . . . . . . . . . . . 12 (𝑅 Fr 𝐵 ↔ ∀𝑐((𝑐𝐵𝑐 ≠ ∅) → ∃𝑦𝑐 Pred(𝑅, 𝑐, 𝑦) = ∅))
2221biimpi 215 . . . . . . . . . . 11 (𝑅 Fr 𝐵 → ∀𝑐((𝑐𝐵𝑐 ≠ ∅) → ∃𝑦𝑐 Pred(𝑅, 𝑐, 𝑦) = ∅))
23 ttrclse 9722 . . . . . . . . . . . . 13 (𝑅 Se 𝐵 → t++(𝑅𝐵) Se 𝐵)
24 setlikespec 6327 . . . . . . . . . . . . 13 ((𝑏𝐵 ∧ t++(𝑅𝐵) Se 𝐵) → Pred(t++(𝑅𝐵), 𝐵, 𝑏) ∈ V)
2523, 24sylan2 594 . . . . . . . . . . . 12 ((𝑏𝐵𝑅 Se 𝐵) → Pred(t++(𝑅𝐵), 𝐵, 𝑏) ∈ V)
2625ancoms 460 . . . . . . . . . . 11 ((𝑅 Se 𝐵𝑏𝐵) → Pred(t++(𝑅𝐵), 𝐵, 𝑏) ∈ V)
27 sseq1 4008 . . . . . . . . . . . . . . 15 (𝑐 = Pred(t++(𝑅𝐵), 𝐵, 𝑏) → (𝑐𝐵 ↔ Pred(t++(𝑅𝐵), 𝐵, 𝑏) ⊆ 𝐵))
28 neeq1 3004 . . . . . . . . . . . . . . 15 (𝑐 = Pred(t++(𝑅𝐵), 𝐵, 𝑏) → (𝑐 ≠ ∅ ↔ Pred(t++(𝑅𝐵), 𝐵, 𝑏) ≠ ∅))
2927, 28anbi12d 632 . . . . . . . . . . . . . 14 (𝑐 = Pred(t++(𝑅𝐵), 𝐵, 𝑏) → ((𝑐𝐵𝑐 ≠ ∅) ↔ (Pred(t++(𝑅𝐵), 𝐵, 𝑏) ⊆ 𝐵 ∧ Pred(t++(𝑅𝐵), 𝐵, 𝑏) ≠ ∅)))
30 predeq2 6304 . . . . . . . . . . . . . . . 16 (𝑐 = Pred(t++(𝑅𝐵), 𝐵, 𝑏) → Pred(𝑅, 𝑐, 𝑦) = Pred(𝑅, Pred(t++(𝑅𝐵), 𝐵, 𝑏), 𝑦))
3130eqeq1d 2735 . . . . . . . . . . . . . . 15 (𝑐 = Pred(t++(𝑅𝐵), 𝐵, 𝑏) → (Pred(𝑅, 𝑐, 𝑦) = ∅ ↔ Pred(𝑅, Pred(t++(𝑅𝐵), 𝐵, 𝑏), 𝑦) = ∅))
3231rexeqbi1dv 3335 . . . . . . . . . . . . . 14 (𝑐 = Pred(t++(𝑅𝐵), 𝐵, 𝑏) → (∃𝑦𝑐 Pred(𝑅, 𝑐, 𝑦) = ∅ ↔ ∃𝑦 ∈ Pred (t++(𝑅𝐵), 𝐵, 𝑏)Pred(𝑅, Pred(t++(𝑅𝐵), 𝐵, 𝑏), 𝑦) = ∅))
3329, 32imbi12d 345 . . . . . . . . . . . . 13 (𝑐 = Pred(t++(𝑅𝐵), 𝐵, 𝑏) → (((𝑐𝐵𝑐 ≠ ∅) → ∃𝑦𝑐 Pred(𝑅, 𝑐, 𝑦) = ∅) ↔ ((Pred(t++(𝑅𝐵), 𝐵, 𝑏) ⊆ 𝐵 ∧ Pred(t++(𝑅𝐵), 𝐵, 𝑏) ≠ ∅) → ∃𝑦 ∈ Pred (t++(𝑅𝐵), 𝐵, 𝑏)Pred(𝑅, Pred(t++(𝑅𝐵), 𝐵, 𝑏), 𝑦) = ∅)))
3433spcgv 3587 . . . . . . . . . . . 12 (Pred(t++(𝑅𝐵), 𝐵, 𝑏) ∈ V → (∀𝑐((𝑐𝐵𝑐 ≠ ∅) → ∃𝑦𝑐 Pred(𝑅, 𝑐, 𝑦) = ∅) → ((Pred(t++(𝑅𝐵), 𝐵, 𝑏) ⊆ 𝐵 ∧ Pred(t++(𝑅𝐵), 𝐵, 𝑏) ≠ ∅) → ∃𝑦 ∈ Pred (t++(𝑅𝐵), 𝐵, 𝑏)Pred(𝑅, Pred(t++(𝑅𝐵), 𝐵, 𝑏), 𝑦) = ∅)))
3534impcom 409 . . . . . . . . . . 11 ((∀𝑐((𝑐𝐵𝑐 ≠ ∅) → ∃𝑦𝑐 Pred(𝑅, 𝑐, 𝑦) = ∅) ∧ Pred(t++(𝑅𝐵), 𝐵, 𝑏) ∈ V) → ((Pred(t++(𝑅𝐵), 𝐵, 𝑏) ⊆ 𝐵 ∧ Pred(t++(𝑅𝐵), 𝐵, 𝑏) ≠ ∅) → ∃𝑦 ∈ Pred (t++(𝑅𝐵), 𝐵, 𝑏)Pred(𝑅, Pred(t++(𝑅𝐵), 𝐵, 𝑏), 𝑦) = ∅))
3622, 26, 35syl2an 597 . . . . . . . . . 10 ((𝑅 Fr 𝐵 ∧ (𝑅 Se 𝐵𝑏𝐵)) → ((Pred(t++(𝑅𝐵), 𝐵, 𝑏) ⊆ 𝐵 ∧ Pred(t++(𝑅𝐵), 𝐵, 𝑏) ≠ ∅) → ∃𝑦 ∈ Pred (t++(𝑅𝐵), 𝐵, 𝑏)Pred(𝑅, Pred(t++(𝑅𝐵), 𝐵, 𝑏), 𝑦) = ∅))
3736anassrs 469 . . . . . . . . 9 (((𝑅 Fr 𝐵𝑅 Se 𝐵) ∧ 𝑏𝐵) → ((Pred(t++(𝑅𝐵), 𝐵, 𝑏) ⊆ 𝐵 ∧ Pred(t++(𝑅𝐵), 𝐵, 𝑏) ≠ ∅) → ∃𝑦 ∈ Pred (t++(𝑅𝐵), 𝐵, 𝑏)Pred(𝑅, Pred(t++(𝑅𝐵), 𝐵, 𝑏), 𝑦) = ∅))
38 predres 6341 . . . . . . . . . . . . . . . . 17 Pred(𝑅, 𝐵, 𝑦) = Pred((𝑅𝐵), 𝐵, 𝑦)
39 predrelss 6339 . . . . . . . . . . . . . . . . . 18 ((𝑅𝐵) ⊆ t++(𝑅𝐵) → Pred((𝑅𝐵), 𝐵, 𝑦) ⊆ Pred(t++(𝑅𝐵), 𝐵, 𝑦))
4013, 39ax-mp 5 . . . . . . . . . . . . . . . . 17 Pred((𝑅𝐵), 𝐵, 𝑦) ⊆ Pred(t++(𝑅𝐵), 𝐵, 𝑦)
4138, 40eqsstri 4017 . . . . . . . . . . . . . . . 16 Pred(𝑅, 𝐵, 𝑦) ⊆ Pred(t++(𝑅𝐵), 𝐵, 𝑦)
42 inss1 4229 . . . . . . . . . . . . . . . . . . . 20 (t++(𝑅𝐵) ∩ (𝐵 × 𝐵)) ⊆ t++(𝑅𝐵)
43 coss1 5856 . . . . . . . . . . . . . . . . . . . 20 ((t++(𝑅𝐵) ∩ (𝐵 × 𝐵)) ⊆ t++(𝑅𝐵) → ((t++(𝑅𝐵) ∩ (𝐵 × 𝐵)) ∘ (t++(𝑅𝐵) ∩ (𝐵 × 𝐵))) ⊆ (t++(𝑅𝐵) ∘ (t++(𝑅𝐵) ∩ (𝐵 × 𝐵))))
4442, 43ax-mp 5 . . . . . . . . . . . . . . . . . . 19 ((t++(𝑅𝐵) ∩ (𝐵 × 𝐵)) ∘ (t++(𝑅𝐵) ∩ (𝐵 × 𝐵))) ⊆ (t++(𝑅𝐵) ∘ (t++(𝑅𝐵) ∩ (𝐵 × 𝐵)))
45 coss2 5857 . . . . . . . . . . . . . . . . . . . 20 ((t++(𝑅𝐵) ∩ (𝐵 × 𝐵)) ⊆ t++(𝑅𝐵) → (t++(𝑅𝐵) ∘ (t++(𝑅𝐵) ∩ (𝐵 × 𝐵))) ⊆ (t++(𝑅𝐵) ∘ t++(𝑅𝐵)))
4642, 45ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (t++(𝑅𝐵) ∘ (t++(𝑅𝐵) ∩ (𝐵 × 𝐵))) ⊆ (t++(𝑅𝐵) ∘ t++(𝑅𝐵))
4744, 46sstri 3992 . . . . . . . . . . . . . . . . . 18 ((t++(𝑅𝐵) ∩ (𝐵 × 𝐵)) ∘ (t++(𝑅𝐵) ∩ (𝐵 × 𝐵))) ⊆ (t++(𝑅𝐵) ∘ t++(𝑅𝐵))
48 ttrcltr 9711 . . . . . . . . . . . . . . . . . 18 (t++(𝑅𝐵) ∘ t++(𝑅𝐵)) ⊆ t++(𝑅𝐵)
4947, 48sstri 3992 . . . . . . . . . . . . . . . . 17 ((t++(𝑅𝐵) ∩ (𝐵 × 𝐵)) ∘ (t++(𝑅𝐵) ∩ (𝐵 × 𝐵))) ⊆ t++(𝑅𝐵)
50 predtrss 6324 . . . . . . . . . . . . . . . . 17 ((((t++(𝑅𝐵) ∩ (𝐵 × 𝐵)) ∘ (t++(𝑅𝐵) ∩ (𝐵 × 𝐵))) ⊆ t++(𝑅𝐵) ∧ 𝑦 ∈ Pred(t++(𝑅𝐵), 𝐵, 𝑏) ∧ 𝑏𝐵) → Pred(t++(𝑅𝐵), 𝐵, 𝑦) ⊆ Pred(t++(𝑅𝐵), 𝐵, 𝑏))
5149, 50mp3an1 1449 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ Pred(t++(𝑅𝐵), 𝐵, 𝑏) ∧ 𝑏𝐵) → Pred(t++(𝑅𝐵), 𝐵, 𝑦) ⊆ Pred(t++(𝑅𝐵), 𝐵, 𝑏))
5241, 51sstrid 3994 . . . . . . . . . . . . . . 15 ((𝑦 ∈ Pred(t++(𝑅𝐵), 𝐵, 𝑏) ∧ 𝑏𝐵) → Pred(𝑅, 𝐵, 𝑦) ⊆ Pred(t++(𝑅𝐵), 𝐵, 𝑏))
53 sspred 6310 . . . . . . . . . . . . . . 15 ((Pred(t++(𝑅𝐵), 𝐵, 𝑏) ⊆ 𝐵 ∧ Pred(𝑅, 𝐵, 𝑦) ⊆ Pred(t++(𝑅𝐵), 𝐵, 𝑏)) → Pred(𝑅, 𝐵, 𝑦) = Pred(𝑅, Pred(t++(𝑅𝐵), 𝐵, 𝑏), 𝑦))
5419, 52, 53sylancr 588 . . . . . . . . . . . . . 14 ((𝑦 ∈ Pred(t++(𝑅𝐵), 𝐵, 𝑏) ∧ 𝑏𝐵) → Pred(𝑅, 𝐵, 𝑦) = Pred(𝑅, Pred(t++(𝑅𝐵), 𝐵, 𝑏), 𝑦))
5554ancoms 460 . . . . . . . . . . . . 13 ((𝑏𝐵𝑦 ∈ Pred(t++(𝑅𝐵), 𝐵, 𝑏)) → Pred(𝑅, 𝐵, 𝑦) = Pred(𝑅, Pred(t++(𝑅𝐵), 𝐵, 𝑏), 𝑦))
5655eqeq1d 2735 . . . . . . . . . . . 12 ((𝑏𝐵𝑦 ∈ Pred(t++(𝑅𝐵), 𝐵, 𝑏)) → (Pred(𝑅, 𝐵, 𝑦) = ∅ ↔ Pred(𝑅, Pred(t++(𝑅𝐵), 𝐵, 𝑏), 𝑦) = ∅))
5756rexbidva 3177 . . . . . . . . . . 11 (𝑏𝐵 → (∃𝑦 ∈ Pred (t++(𝑅𝐵), 𝐵, 𝑏)Pred(𝑅, 𝐵, 𝑦) = ∅ ↔ ∃𝑦 ∈ Pred (t++(𝑅𝐵), 𝐵, 𝑏)Pred(𝑅, Pred(t++(𝑅𝐵), 𝐵, 𝑏), 𝑦) = ∅))
58 ssrexv 4052 . . . . . . . . . . . 12 (Pred(t++(𝑅𝐵), 𝐵, 𝑏) ⊆ 𝐵 → (∃𝑦 ∈ Pred (t++(𝑅𝐵), 𝐵, 𝑏)Pred(𝑅, 𝐵, 𝑦) = ∅ → ∃𝑦𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅))
5919, 58ax-mp 5 . . . . . . . . . . 11 (∃𝑦 ∈ Pred (t++(𝑅𝐵), 𝐵, 𝑏)Pred(𝑅, 𝐵, 𝑦) = ∅ → ∃𝑦𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅)
6057, 59syl6bir 254 . . . . . . . . . 10 (𝑏𝐵 → (∃𝑦 ∈ Pred (t++(𝑅𝐵), 𝐵, 𝑏)Pred(𝑅, Pred(t++(𝑅𝐵), 𝐵, 𝑏), 𝑦) = ∅ → ∃𝑦𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅))
6160adantl 483 . . . . . . . . 9 (((𝑅 Fr 𝐵𝑅 Se 𝐵) ∧ 𝑏𝐵) → (∃𝑦 ∈ Pred (t++(𝑅𝐵), 𝐵, 𝑏)Pred(𝑅, Pred(t++(𝑅𝐵), 𝐵, 𝑏), 𝑦) = ∅ → ∃𝑦𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅))
6237, 61syld 47 . . . . . . . 8 (((𝑅 Fr 𝐵𝑅 Se 𝐵) ∧ 𝑏𝐵) → ((Pred(t++(𝑅𝐵), 𝐵, 𝑏) ⊆ 𝐵 ∧ Pred(t++(𝑅𝐵), 𝐵, 𝑏) ≠ ∅) → ∃𝑦𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅))
6320, 62syl5 34 . . . . . . 7 (((𝑅 Fr 𝐵𝑅 Se 𝐵) ∧ 𝑏𝐵) → (Pred(𝑅, 𝐵, 𝑏) ≠ ∅ → ∃𝑦𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅))
649, 63pm2.61dne 3029 . . . . . 6 (((𝑅 Fr 𝐵𝑅 Se 𝐵) ∧ 𝑏𝐵) → ∃𝑦𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅)
6564ex 414 . . . . 5 ((𝑅 Fr 𝐵𝑅 Se 𝐵) → (𝑏𝐵 → ∃𝑦𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅))
6665exlimdv 1937 . . . 4 ((𝑅 Fr 𝐵𝑅 Se 𝐵) → (∃𝑏 𝑏𝐵 → ∃𝑦𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅))
674, 66biimtrid 241 . . 3 ((𝑅 Fr 𝐵𝑅 Se 𝐵) → (𝐵 ≠ ∅ → ∃𝑦𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅))
683, 67syl6com 37 . 2 ((𝑅 Fr 𝐴𝑅 Se 𝐴) → (𝐵𝐴 → (𝐵 ≠ ∅ → ∃𝑦𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅)))
6968imp32 420 1 (((𝑅 Fr 𝐴𝑅 Se 𝐴) ∧ (𝐵𝐴𝐵 ≠ ∅)) → ∃𝑦𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wal 1540   = wceq 1542  wex 1782  wcel 2107  wne 2941  wrex 3071  Vcvv 3475  cin 3948  wss 3949  c0 4323   Fr wfr 5629   Se wse 5630   × cxp 5675  cres 5679  ccom 5681  Rel wrel 5682  Predcpred 6300  t++cttrcl 9702
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725  ax-inf2 9636
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-oadd 8470  df-ttrcl 9703
This theorem is referenced by:  frind  9745  frr1  9754
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