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Theorem frmin 9690
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 6302 and tz7.5 6339. (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 5601 . . . 4 (𝐵𝐴 → (𝑅 Fr 𝐴𝑅 Fr 𝐵))
2 sess2 5603 . . . 4 (𝐵𝐴 → (𝑅 Se 𝐴𝑅 Se 𝐵))
31, 2anim12d 610 . . 3 (𝐵𝐴 → ((𝑅 Fr 𝐴𝑅 Se 𝐴) → (𝑅 Fr 𝐵𝑅 Se 𝐵)))
4 n0 4307 . . . 4 (𝐵 ≠ ∅ ↔ ∃𝑏 𝑏𝐵)
5 predeq3 6258 . . . . . . . . . . 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 3580 . . . . . . . . 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 6294 . . . . . . . . . . 11 Pred(𝑅, 𝐵, 𝑏) = Pred((𝑅𝐵), 𝐵, 𝑏)
11 relres 5967 . . . . . . . . . . . . 13 Rel (𝑅𝐵)
12 ssttrcl 9656 . . . . . . . . . . . . 13 (Rel (𝑅𝐵) → (𝑅𝐵) ⊆ t++(𝑅𝐵))
1311, 12ax-mp 5 . . . . . . . . . . . 12 (𝑅𝐵) ⊆ t++(𝑅𝐵)
14 predrelss 6292 . . . . . . . . . . . 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 3979 . . . . . . . . . 10 Pred(𝑅, 𝐵, 𝑏) ⊆ Pred(t++(𝑅𝐵), 𝐵, 𝑏)
17 ssn0 4361 . . . . . . . . . 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 6262 . . . . . . . . 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 6274 . . . . . . . . . . . 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 9668 . . . . . . . . . . . . 13 (𝑅 Se 𝐵 → t++(𝑅𝐵) Se 𝐵)
24 setlikespec 6280 . . . . . . . . . . . . 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 3970 . . . . . . . . . . . . . . 15 (𝑐 = Pred(t++(𝑅𝐵), 𝐵, 𝑏) → (𝑐𝐵 ↔ Pred(t++(𝑅𝐵), 𝐵, 𝑏) ⊆ 𝐵))
28 neeq1 3003 . . . . . . . . . . . . . . 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 6257 . . . . . . . . . . . . . . . 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 3307 . . . . . . . . . . . . . 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 3554 . . . . . . . . . . . 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 6294 . . . . . . . . . . . . . . . . 17 Pred(𝑅, 𝐵, 𝑦) = Pred((𝑅𝐵), 𝐵, 𝑦)
39 predrelss 6292 . . . . . . . . . . . . . . . . . 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 3979 . . . . . . . . . . . . . . . 16 Pred(𝑅, 𝐵, 𝑦) ⊆ Pred(t++(𝑅𝐵), 𝐵, 𝑦)
42 inss1 4189 . . . . . . . . . . . . . . . . . . . 20 (t++(𝑅𝐵) ∩ (𝐵 × 𝐵)) ⊆ t++(𝑅𝐵)
43 coss1 5812 . . . . . . . . . . . . . . . . . . . 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 5813 . . . . . . . . . . . . . . . . . . . 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 3954 . . . . . . . . . . . . . . . . . 18 ((t++(𝑅𝐵) ∩ (𝐵 × 𝐵)) ∘ (t++(𝑅𝐵) ∩ (𝐵 × 𝐵))) ⊆ (t++(𝑅𝐵) ∘ t++(𝑅𝐵))
48 ttrcltr 9657 . . . . . . . . . . . . . . . . . 18 (t++(𝑅𝐵) ∘ t++(𝑅𝐵)) ⊆ t++(𝑅𝐵)
4947, 48sstri 3954 . . . . . . . . . . . . . . . . 17 ((t++(𝑅𝐵) ∩ (𝐵 × 𝐵)) ∘ (t++(𝑅𝐵) ∩ (𝐵 × 𝐵))) ⊆ t++(𝑅𝐵)
50 predtrss 6277 . . . . . . . . . . . . . . . . 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 3956 . . . . . . . . . . . . . . 15 ((𝑦 ∈ Pred(t++(𝑅𝐵), 𝐵, 𝑏) ∧ 𝑏𝐵) → Pred(𝑅, 𝐵, 𝑦) ⊆ Pred(t++(𝑅𝐵), 𝐵, 𝑏))
53 sspred 6263 . . . . . . . . . . . . . . 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 3170 . . . . . . . . . . 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 4012 . . . . . . . . . . . 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 3028 . . . . . 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 2940  wrex 3070  Vcvv 3444  cin 3910  wss 3911  c0 4283   Fr wfr 5586   Se wse 5587   × cxp 5632  cres 5636  ccom 5638  Rel wrel 5639  Predcpred 6253  t++cttrcl 9648
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 5243  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673  ax-inf2 9582
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 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-oadd 8417  df-ttrcl 9649
This theorem is referenced by:  frind  9691  frr1  9700
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