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Theorem ltsosr 11109
Description: Signed real 'less than' is a strict ordering. (Contributed by NM, 19-Feb-1996.) (New usage is discouraged.)
Assertion
Ref Expression
ltsosr <R Or R

Proof of Theorem ltsosr
Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 𝑢 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nr 11071 . . 3 R = ((P × P) / ~R )
2 breq1 5145 . . . 4 ([⟨𝑥, 𝑦⟩] ~R = 𝑓 → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R𝑓 <R [⟨𝑧, 𝑤⟩] ~R ))
3 eqeq1 2731 . . . . . 6 ([⟨𝑥, 𝑦⟩] ~R = 𝑓 → ([⟨𝑥, 𝑦⟩] ~R = [⟨𝑧, 𝑤⟩] ~R𝑓 = [⟨𝑧, 𝑤⟩] ~R ))
4 breq2 5146 . . . . . 6 ([⟨𝑥, 𝑦⟩] ~R = 𝑓 → ([⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R ↔ [⟨𝑧, 𝑤⟩] ~R <R 𝑓))
53, 4orbi12d 917 . . . . 5 ([⟨𝑥, 𝑦⟩] ~R = 𝑓 → (([⟨𝑥, 𝑦⟩] ~R = [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R ) ↔ (𝑓 = [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R 𝑓)))
65notbid 318 . . . 4 ([⟨𝑥, 𝑦⟩] ~R = 𝑓 → (¬ ([⟨𝑥, 𝑦⟩] ~R = [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R ) ↔ ¬ (𝑓 = [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R 𝑓)))
72, 6bibi12d 345 . . 3 ([⟨𝑥, 𝑦⟩] ~R = 𝑓 → (([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ↔ ¬ ([⟨𝑥, 𝑦⟩] ~R = [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R )) ↔ (𝑓 <R [⟨𝑧, 𝑤⟩] ~R ↔ ¬ (𝑓 = [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R 𝑓))))
8 breq2 5146 . . . 4 ([⟨𝑧, 𝑤⟩] ~R = 𝑔 → (𝑓 <R [⟨𝑧, 𝑤⟩] ~R𝑓 <R 𝑔))
9 eqeq2 2739 . . . . . 6 ([⟨𝑧, 𝑤⟩] ~R = 𝑔 → (𝑓 = [⟨𝑧, 𝑤⟩] ~R𝑓 = 𝑔))
10 breq1 5145 . . . . . 6 ([⟨𝑧, 𝑤⟩] ~R = 𝑔 → ([⟨𝑧, 𝑤⟩] ~R <R 𝑓𝑔 <R 𝑓))
119, 10orbi12d 917 . . . . 5 ([⟨𝑧, 𝑤⟩] ~R = 𝑔 → ((𝑓 = [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R 𝑓) ↔ (𝑓 = 𝑔𝑔 <R 𝑓)))
1211notbid 318 . . . 4 ([⟨𝑧, 𝑤⟩] ~R = 𝑔 → (¬ (𝑓 = [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R 𝑓) ↔ ¬ (𝑓 = 𝑔𝑔 <R 𝑓)))
138, 12bibi12d 345 . . 3 ([⟨𝑧, 𝑤⟩] ~R = 𝑔 → ((𝑓 <R [⟨𝑧, 𝑤⟩] ~R ↔ ¬ (𝑓 = [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R 𝑓)) ↔ (𝑓 <R 𝑔 ↔ ¬ (𝑓 = 𝑔𝑔 <R 𝑓))))
14 ltsrpr 11092 . . . 4 ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ↔ (𝑥 +P 𝑤)<P (𝑦 +P 𝑧))
15 addclpr 11033 . . . . . . 7 ((𝑥P𝑤P) → (𝑥 +P 𝑤) ∈ P)
16 addclpr 11033 . . . . . . 7 ((𝑦P𝑧P) → (𝑦 +P 𝑧) ∈ P)
17 ltsopr 11047 . . . . . . . 8 <P Or P
18 sotric 5612 . . . . . . . 8 ((<P Or P ∧ ((𝑥 +P 𝑤) ∈ P ∧ (𝑦 +P 𝑧) ∈ P)) → ((𝑥 +P 𝑤)<P (𝑦 +P 𝑧) ↔ ¬ ((𝑥 +P 𝑤) = (𝑦 +P 𝑧) ∨ (𝑦 +P 𝑧)<P (𝑥 +P 𝑤))))
1917, 18mpan 689 . . . . . . 7 (((𝑥 +P 𝑤) ∈ P ∧ (𝑦 +P 𝑧) ∈ P) → ((𝑥 +P 𝑤)<P (𝑦 +P 𝑧) ↔ ¬ ((𝑥 +P 𝑤) = (𝑦 +P 𝑧) ∨ (𝑦 +P 𝑧)<P (𝑥 +P 𝑤))))
2015, 16, 19syl2an 595 . . . . . 6 (((𝑥P𝑤P) ∧ (𝑦P𝑧P)) → ((𝑥 +P 𝑤)<P (𝑦 +P 𝑧) ↔ ¬ ((𝑥 +P 𝑤) = (𝑦 +P 𝑧) ∨ (𝑦 +P 𝑧)<P (𝑥 +P 𝑤))))
2120an42s 660 . . . . 5 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ((𝑥 +P 𝑤)<P (𝑦 +P 𝑧) ↔ ¬ ((𝑥 +P 𝑤) = (𝑦 +P 𝑧) ∨ (𝑦 +P 𝑧)<P (𝑥 +P 𝑤))))
22 enreceq 11081 . . . . . . 7 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ([⟨𝑥, 𝑦⟩] ~R = [⟨𝑧, 𝑤⟩] ~R ↔ (𝑥 +P 𝑤) = (𝑦 +P 𝑧)))
23 ltsrpr 11092 . . . . . . . . 9 ([⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R ↔ (𝑧 +P 𝑦)<P (𝑤 +P 𝑥))
24 addcompr 11036 . . . . . . . . . 10 (𝑧 +P 𝑦) = (𝑦 +P 𝑧)
25 addcompr 11036 . . . . . . . . . 10 (𝑤 +P 𝑥) = (𝑥 +P 𝑤)
2624, 25breq12i 5151 . . . . . . . . 9 ((𝑧 +P 𝑦)<P (𝑤 +P 𝑥) ↔ (𝑦 +P 𝑧)<P (𝑥 +P 𝑤))
2723, 26bitri 275 . . . . . . . 8 ([⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R ↔ (𝑦 +P 𝑧)<P (𝑥 +P 𝑤))
2827a1i 11 . . . . . . 7 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ([⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R ↔ (𝑦 +P 𝑧)<P (𝑥 +P 𝑤)))
2922, 28orbi12d 917 . . . . . 6 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → (([⟨𝑥, 𝑦⟩] ~R = [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R ) ↔ ((𝑥 +P 𝑤) = (𝑦 +P 𝑧) ∨ (𝑦 +P 𝑧)<P (𝑥 +P 𝑤))))
3029notbid 318 . . . . 5 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → (¬ ([⟨𝑥, 𝑦⟩] ~R = [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R ) ↔ ¬ ((𝑥 +P 𝑤) = (𝑦 +P 𝑧) ∨ (𝑦 +P 𝑧)<P (𝑥 +P 𝑤))))
3121, 30bitr4d 282 . . . 4 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ((𝑥 +P 𝑤)<P (𝑦 +P 𝑧) ↔ ¬ ([⟨𝑥, 𝑦⟩] ~R = [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R )))
3214, 31bitrid 283 . . 3 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ↔ ¬ ([⟨𝑥, 𝑦⟩] ~R = [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R )))
331, 7, 13, 322ecoptocl 8818 . 2 ((𝑓R𝑔R) → (𝑓 <R 𝑔 ↔ ¬ (𝑓 = 𝑔𝑔 <R 𝑓)))
342anbi1d 629 . . . 4 ([⟨𝑥, 𝑦⟩] ~R = 𝑓 → (([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ∧ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ) ↔ (𝑓 <R [⟨𝑧, 𝑤⟩] ~R ∧ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R )))
35 breq1 5145 . . . 4 ([⟨𝑥, 𝑦⟩] ~R = 𝑓 → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R𝑓 <R [⟨𝑣, 𝑢⟩] ~R ))
3634, 35imbi12d 344 . . 3 ([⟨𝑥, 𝑦⟩] ~R = 𝑓 → ((([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ∧ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ) → [⟨𝑥, 𝑦⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ) ↔ ((𝑓 <R [⟨𝑧, 𝑤⟩] ~R ∧ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ) → 𝑓 <R [⟨𝑣, 𝑢⟩] ~R )))
37 breq1 5145 . . . . 5 ([⟨𝑧, 𝑤⟩] ~R = 𝑔 → ([⟨𝑧, 𝑤⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R𝑔 <R [⟨𝑣, 𝑢⟩] ~R ))
388, 37anbi12d 630 . . . 4 ([⟨𝑧, 𝑤⟩] ~R = 𝑔 → ((𝑓 <R [⟨𝑧, 𝑤⟩] ~R ∧ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ) ↔ (𝑓 <R 𝑔𝑔 <R [⟨𝑣, 𝑢⟩] ~R )))
3938imbi1d 341 . . 3 ([⟨𝑧, 𝑤⟩] ~R = 𝑔 → (((𝑓 <R [⟨𝑧, 𝑤⟩] ~R ∧ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ) → 𝑓 <R [⟨𝑣, 𝑢⟩] ~R ) ↔ ((𝑓 <R 𝑔𝑔 <R [⟨𝑣, 𝑢⟩] ~R ) → 𝑓 <R [⟨𝑣, 𝑢⟩] ~R )))
40 breq2 5146 . . . . 5 ([⟨𝑣, 𝑢⟩] ~R = → (𝑔 <R [⟨𝑣, 𝑢⟩] ~R𝑔 <R ))
4140anbi2d 628 . . . 4 ([⟨𝑣, 𝑢⟩] ~R = → ((𝑓 <R 𝑔𝑔 <R [⟨𝑣, 𝑢⟩] ~R ) ↔ (𝑓 <R 𝑔𝑔 <R )))
42 breq2 5146 . . . 4 ([⟨𝑣, 𝑢⟩] ~R = → (𝑓 <R [⟨𝑣, 𝑢⟩] ~R𝑓 <R ))
4341, 42imbi12d 344 . . 3 ([⟨𝑣, 𝑢⟩] ~R = → (((𝑓 <R 𝑔𝑔 <R [⟨𝑣, 𝑢⟩] ~R ) → 𝑓 <R [⟨𝑣, 𝑢⟩] ~R ) ↔ ((𝑓 <R 𝑔𝑔 <R ) → 𝑓 <R )))
44 ovex 7447 . . . . . . . . . 10 (𝑥 +P 𝑤) ∈ V
45 ovex 7447 . . . . . . . . . 10 (𝑦 +P 𝑧) ∈ V
46 ltapr 11060 . . . . . . . . . 10 (P → (𝑓<P 𝑔 ↔ ( +P 𝑓)<P ( +P 𝑔)))
47 vex 3473 . . . . . . . . . 10 𝑢 ∈ V
48 addcompr 11036 . . . . . . . . . 10 (𝑓 +P 𝑔) = (𝑔 +P 𝑓)
4944, 45, 46, 47, 48caovord2 7627 . . . . . . . . 9 (𝑢P → ((𝑥 +P 𝑤)<P (𝑦 +P 𝑧) ↔ ((𝑥 +P 𝑤) +P 𝑢)<P ((𝑦 +P 𝑧) +P 𝑢)))
50 addasspr 11037 . . . . . . . . . 10 ((𝑥 +P 𝑤) +P 𝑢) = (𝑥 +P (𝑤 +P 𝑢))
51 addasspr 11037 . . . . . . . . . 10 ((𝑦 +P 𝑧) +P 𝑢) = (𝑦 +P (𝑧 +P 𝑢))
5250, 51breq12i 5151 . . . . . . . . 9 (((𝑥 +P 𝑤) +P 𝑢)<P ((𝑦 +P 𝑧) +P 𝑢) ↔ (𝑥 +P (𝑤 +P 𝑢))<P (𝑦 +P (𝑧 +P 𝑢)))
5349, 52bitrdi 287 . . . . . . . 8 (𝑢P → ((𝑥 +P 𝑤)<P (𝑦 +P 𝑧) ↔ (𝑥 +P (𝑤 +P 𝑢))<P (𝑦 +P (𝑧 +P 𝑢))))
5414, 53bitrid 283 . . . . . . 7 (𝑢P → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ↔ (𝑥 +P (𝑤 +P 𝑢))<P (𝑦 +P (𝑧 +P 𝑢))))
55 ltsrpr 11092 . . . . . . . 8 ([⟨𝑧, 𝑤⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ↔ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣))
56 ltapr 11060 . . . . . . . 8 (𝑦P → ((𝑧 +P 𝑢)<P (𝑤 +P 𝑣) ↔ (𝑦 +P (𝑧 +P 𝑢))<P (𝑦 +P (𝑤 +P 𝑣))))
5755, 56bitrid 283 . . . . . . 7 (𝑦P → ([⟨𝑧, 𝑤⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ↔ (𝑦 +P (𝑧 +P 𝑢))<P (𝑦 +P (𝑤 +P 𝑣))))
5854, 57bi2anan9r 638 . . . . . 6 ((𝑦P𝑢P) → (([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ∧ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ) ↔ ((𝑥 +P (𝑤 +P 𝑢))<P (𝑦 +P (𝑧 +P 𝑢)) ∧ (𝑦 +P (𝑧 +P 𝑢))<P (𝑦 +P (𝑤 +P 𝑣)))))
59 ltrelpr 11013 . . . . . . . 8 <P ⊆ (P × P)
6017, 59sotri 6127 . . . . . . 7 (((𝑥 +P (𝑤 +P 𝑢))<P (𝑦 +P (𝑧 +P 𝑢)) ∧ (𝑦 +P (𝑧 +P 𝑢))<P (𝑦 +P (𝑤 +P 𝑣))) → (𝑥 +P (𝑤 +P 𝑢))<P (𝑦 +P (𝑤 +P 𝑣)))
61 dmplp 11027 . . . . . . . . 9 dom +P = (P × P)
62 0npr 11007 . . . . . . . . 9 ¬ ∅ ∈ P
63 ltapr 11060 . . . . . . . . 9 (𝑤P → ((𝑥 +P 𝑢)<P (𝑦 +P 𝑣) ↔ (𝑤 +P (𝑥 +P 𝑢))<P (𝑤 +P (𝑦 +P 𝑣))))
6461, 59, 62, 63ndmovordi 7606 . . . . . . . 8 ((𝑤 +P (𝑥 +P 𝑢))<P (𝑤 +P (𝑦 +P 𝑣)) → (𝑥 +P 𝑢)<P (𝑦 +P 𝑣))
65 vex 3473 . . . . . . . . . 10 𝑥 ∈ V
66 vex 3473 . . . . . . . . . 10 𝑤 ∈ V
67 addasspr 11037 . . . . . . . . . 10 ((𝑓 +P 𝑔) +P ) = (𝑓 +P (𝑔 +P ))
6865, 66, 47, 48, 67caov12 7643 . . . . . . . . 9 (𝑥 +P (𝑤 +P 𝑢)) = (𝑤 +P (𝑥 +P 𝑢))
69 vex 3473 . . . . . . . . . 10 𝑦 ∈ V
70 vex 3473 . . . . . . . . . 10 𝑣 ∈ V
7169, 66, 70, 48, 67caov12 7643 . . . . . . . . 9 (𝑦 +P (𝑤 +P 𝑣)) = (𝑤 +P (𝑦 +P 𝑣))
7268, 71breq12i 5151 . . . . . . . 8 ((𝑥 +P (𝑤 +P 𝑢))<P (𝑦 +P (𝑤 +P 𝑣)) ↔ (𝑤 +P (𝑥 +P 𝑢))<P (𝑤 +P (𝑦 +P 𝑣)))
73 ltsrpr 11092 . . . . . . . 8 ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ↔ (𝑥 +P 𝑢)<P (𝑦 +P 𝑣))
7464, 72, 733imtr4i 292 . . . . . . 7 ((𝑥 +P (𝑤 +P 𝑢))<P (𝑦 +P (𝑤 +P 𝑣)) → [⟨𝑥, 𝑦⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R )
7560, 74syl 17 . . . . . 6 (((𝑥 +P (𝑤 +P 𝑢))<P (𝑦 +P (𝑧 +P 𝑢)) ∧ (𝑦 +P (𝑧 +P 𝑢))<P (𝑦 +P (𝑤 +P 𝑣))) → [⟨𝑥, 𝑦⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R )
7658, 75biimtrdi 252 . . . . 5 ((𝑦P𝑢P) → (([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ∧ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ) → [⟨𝑥, 𝑦⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ))
7776ad2ant2l 745 . . . 4 (((𝑥P𝑦P) ∧ (𝑣P𝑢P)) → (([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ∧ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ) → [⟨𝑥, 𝑦⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ))
78773adant2 1129 . . 3 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑣P𝑢P)) → (([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ∧ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ) → [⟨𝑥, 𝑦⟩] ~R <R [⟨𝑣, 𝑢⟩] ~R ))
791, 36, 39, 43, 783ecoptocl 8819 . 2 ((𝑓R𝑔RR) → ((𝑓 <R 𝑔𝑔 <R ) → 𝑓 <R ))
8033, 79isso2i 5619 1 <R Or R
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  wo 846   = wceq 1534  wcel 2099  cop 4630   class class class wbr 5142   Or wor 5583  (class class class)co 7414  [cec 8716  Pcnp 10874   +P cpp 10876  <P cltp 10878   ~R cer 10879  Rcnr 10880   <R cltr 10886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-inf2 9656
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-oadd 8484  df-omul 8485  df-er 8718  df-ec 8720  df-qs 8724  df-ni 10887  df-pli 10888  df-mi 10889  df-lti 10890  df-plpq 10923  df-mpq 10924  df-ltpq 10925  df-enq 10926  df-nq 10927  df-erq 10928  df-plq 10929  df-mq 10930  df-1nq 10931  df-rq 10932  df-ltnq 10933  df-np 10996  df-plp 10998  df-ltp 11000  df-enr 11070  df-nr 11071  df-ltr 11074
This theorem is referenced by:  1ne0sr  11111  addgt0sr  11119  sqgt0sr  11121  supsrlem  11126  axpre-lttri  11180  axpre-lttrn  11181
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