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Theorem ltsosr 7507
Description: Signed real 'less than' is a strict ordering. (Contributed by NM, 19-Feb-1996.)
Assertion
Ref Expression
ltsosr <R Or R

Proof of Theorem ltsosr
Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 𝑟 𝑠 𝑡 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltposr 7506 . 2 <R Po R
2 df-nr 7470 . . . 4 R = ((P × P) / ~R )
3 breq1 3898 . . . . 5 ([⟨𝑎, 𝑏⟩] ~R = 𝑥 → ([⟨𝑎, 𝑏⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R𝑥 <R [⟨𝑐, 𝑑⟩] ~R ))
4 breq1 3898 . . . . . 6 ([⟨𝑎, 𝑏⟩] ~R = 𝑥 → ([⟨𝑎, 𝑏⟩] ~R <R [⟨𝑒, 𝑓⟩] ~R𝑥 <R [⟨𝑒, 𝑓⟩] ~R ))
54orbi1d 763 . . . . 5 ([⟨𝑎, 𝑏⟩] ~R = 𝑥 → (([⟨𝑎, 𝑏⟩] ~R <R [⟨𝑒, 𝑓⟩] ~R ∨ [⟨𝑒, 𝑓⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R ) ↔ (𝑥 <R [⟨𝑒, 𝑓⟩] ~R ∨ [⟨𝑒, 𝑓⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R )))
63, 5imbi12d 233 . . . 4 ([⟨𝑎, 𝑏⟩] ~R = 𝑥 → (([⟨𝑎, 𝑏⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R → ([⟨𝑎, 𝑏⟩] ~R <R [⟨𝑒, 𝑓⟩] ~R ∨ [⟨𝑒, 𝑓⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R )) ↔ (𝑥 <R [⟨𝑐, 𝑑⟩] ~R → (𝑥 <R [⟨𝑒, 𝑓⟩] ~R ∨ [⟨𝑒, 𝑓⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R ))))
7 breq2 3899 . . . . 5 ([⟨𝑐, 𝑑⟩] ~R = 𝑦 → (𝑥 <R [⟨𝑐, 𝑑⟩] ~R𝑥 <R 𝑦))
8 breq2 3899 . . . . . 6 ([⟨𝑐, 𝑑⟩] ~R = 𝑦 → ([⟨𝑒, 𝑓⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R ↔ [⟨𝑒, 𝑓⟩] ~R <R 𝑦))
98orbi2d 762 . . . . 5 ([⟨𝑐, 𝑑⟩] ~R = 𝑦 → ((𝑥 <R [⟨𝑒, 𝑓⟩] ~R ∨ [⟨𝑒, 𝑓⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R ) ↔ (𝑥 <R [⟨𝑒, 𝑓⟩] ~R ∨ [⟨𝑒, 𝑓⟩] ~R <R 𝑦)))
107, 9imbi12d 233 . . . 4 ([⟨𝑐, 𝑑⟩] ~R = 𝑦 → ((𝑥 <R [⟨𝑐, 𝑑⟩] ~R → (𝑥 <R [⟨𝑒, 𝑓⟩] ~R ∨ [⟨𝑒, 𝑓⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R )) ↔ (𝑥 <R 𝑦 → (𝑥 <R [⟨𝑒, 𝑓⟩] ~R ∨ [⟨𝑒, 𝑓⟩] ~R <R 𝑦))))
11 breq2 3899 . . . . . 6 ([⟨𝑒, 𝑓⟩] ~R = 𝑧 → (𝑥 <R [⟨𝑒, 𝑓⟩] ~R𝑥 <R 𝑧))
12 breq1 3898 . . . . . 6 ([⟨𝑒, 𝑓⟩] ~R = 𝑧 → ([⟨𝑒, 𝑓⟩] ~R <R 𝑦𝑧 <R 𝑦))
1311, 12orbi12d 765 . . . . 5 ([⟨𝑒, 𝑓⟩] ~R = 𝑧 → ((𝑥 <R [⟨𝑒, 𝑓⟩] ~R ∨ [⟨𝑒, 𝑓⟩] ~R <R 𝑦) ↔ (𝑥 <R 𝑧𝑧 <R 𝑦)))
1413imbi2d 229 . . . 4 ([⟨𝑒, 𝑓⟩] ~R = 𝑧 → ((𝑥 <R 𝑦 → (𝑥 <R [⟨𝑒, 𝑓⟩] ~R ∨ [⟨𝑒, 𝑓⟩] ~R <R 𝑦)) ↔ (𝑥 <R 𝑦 → (𝑥 <R 𝑧𝑧 <R 𝑦))))
15 simp1l 988 . . . . . . . . 9 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → 𝑎P)
16 simp3r 993 . . . . . . . . 9 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → 𝑓P)
17 addclpr 7293 . . . . . . . . 9 ((𝑎P𝑓P) → (𝑎 +P 𝑓) ∈ P)
1815, 16, 17syl2anc 406 . . . . . . . 8 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → (𝑎 +P 𝑓) ∈ P)
19 simp2r 991 . . . . . . . 8 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → 𝑑P)
20 addclpr 7293 . . . . . . . 8 (((𝑎 +P 𝑓) ∈ P𝑑P) → ((𝑎 +P 𝑓) +P 𝑑) ∈ P)
2118, 19, 20syl2anc 406 . . . . . . 7 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ((𝑎 +P 𝑓) +P 𝑑) ∈ P)
22 simp2l 990 . . . . . . . . 9 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → 𝑐P)
23 addclpr 7293 . . . . . . . . 9 ((𝑓P𝑐P) → (𝑓 +P 𝑐) ∈ P)
2416, 22, 23syl2anc 406 . . . . . . . 8 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → (𝑓 +P 𝑐) ∈ P)
25 simp1r 989 . . . . . . . 8 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → 𝑏P)
26 addclpr 7293 . . . . . . . 8 (((𝑓 +P 𝑐) ∈ P𝑏P) → ((𝑓 +P 𝑐) +P 𝑏) ∈ P)
2724, 25, 26syl2anc 406 . . . . . . 7 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ((𝑓 +P 𝑐) +P 𝑏) ∈ P)
28 simp3l 992 . . . . . . . . 9 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → 𝑒P)
29 addclpr 7293 . . . . . . . . 9 ((𝑏P𝑒P) → (𝑏 +P 𝑒) ∈ P)
3025, 28, 29syl2anc 406 . . . . . . . 8 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → (𝑏 +P 𝑒) ∈ P)
31 addclpr 7293 . . . . . . . 8 (((𝑏 +P 𝑒) ∈ P𝑑P) → ((𝑏 +P 𝑒) +P 𝑑) ∈ P)
3230, 19, 31syl2anc 406 . . . . . . 7 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ((𝑏 +P 𝑒) +P 𝑑) ∈ P)
33 ltsopr 7352 . . . . . . . 8 <P Or P
34 sowlin 4202 . . . . . . . 8 ((<P Or P ∧ (((𝑎 +P 𝑓) +P 𝑑) ∈ P ∧ ((𝑓 +P 𝑐) +P 𝑏) ∈ P ∧ ((𝑏 +P 𝑒) +P 𝑑) ∈ P)) → (((𝑎 +P 𝑓) +P 𝑑)<P ((𝑓 +P 𝑐) +P 𝑏) → (((𝑎 +P 𝑓) +P 𝑑)<P ((𝑏 +P 𝑒) +P 𝑑) ∨ ((𝑏 +P 𝑒) +P 𝑑)<P ((𝑓 +P 𝑐) +P 𝑏))))
3533, 34mpan 418 . . . . . . 7 ((((𝑎 +P 𝑓) +P 𝑑) ∈ P ∧ ((𝑓 +P 𝑐) +P 𝑏) ∈ P ∧ ((𝑏 +P 𝑒) +P 𝑑) ∈ P) → (((𝑎 +P 𝑓) +P 𝑑)<P ((𝑓 +P 𝑐) +P 𝑏) → (((𝑎 +P 𝑓) +P 𝑑)<P ((𝑏 +P 𝑒) +P 𝑑) ∨ ((𝑏 +P 𝑒) +P 𝑑)<P ((𝑓 +P 𝑐) +P 𝑏))))
3621, 27, 32, 35syl3anc 1199 . . . . . 6 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → (((𝑎 +P 𝑓) +P 𝑑)<P ((𝑓 +P 𝑐) +P 𝑏) → (((𝑎 +P 𝑓) +P 𝑑)<P ((𝑏 +P 𝑒) +P 𝑑) ∨ ((𝑏 +P 𝑒) +P 𝑑)<P ((𝑓 +P 𝑐) +P 𝑏))))
37 addclpr 7293 . . . . . . . . 9 ((𝑎P𝑑P) → (𝑎 +P 𝑑) ∈ P)
3815, 19, 37syl2anc 406 . . . . . . . 8 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → (𝑎 +P 𝑑) ∈ P)
39 addclpr 7293 . . . . . . . . 9 ((𝑏P𝑐P) → (𝑏 +P 𝑐) ∈ P)
4025, 22, 39syl2anc 406 . . . . . . . 8 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → (𝑏 +P 𝑐) ∈ P)
41 ltaprg 7375 . . . . . . . 8 (((𝑎 +P 𝑑) ∈ P ∧ (𝑏 +P 𝑐) ∈ P𝑓P) → ((𝑎 +P 𝑑)<P (𝑏 +P 𝑐) ↔ (𝑓 +P (𝑎 +P 𝑑))<P (𝑓 +P (𝑏 +P 𝑐))))
4238, 40, 16, 41syl3anc 1199 . . . . . . 7 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ((𝑎 +P 𝑑)<P (𝑏 +P 𝑐) ↔ (𝑓 +P (𝑎 +P 𝑑))<P (𝑓 +P (𝑏 +P 𝑐))))
43 addcomprg 7334 . . . . . . . . . . 11 ((𝑟P𝑠P) → (𝑟 +P 𝑠) = (𝑠 +P 𝑟))
4443adantl 273 . . . . . . . . . 10 ((((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) ∧ (𝑟P𝑠P)) → (𝑟 +P 𝑠) = (𝑠 +P 𝑟))
45 addassprg 7335 . . . . . . . . . . 11 ((𝑟P𝑠P𝑡P) → ((𝑟 +P 𝑠) +P 𝑡) = (𝑟 +P (𝑠 +P 𝑡)))
4645adantl 273 . . . . . . . . . 10 ((((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) ∧ (𝑟P𝑠P𝑡P)) → ((𝑟 +P 𝑠) +P 𝑡) = (𝑟 +P (𝑠 +P 𝑡)))
4716, 15, 19, 44, 46caov12d 5906 . . . . . . . . 9 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → (𝑓 +P (𝑎 +P 𝑑)) = (𝑎 +P (𝑓 +P 𝑑)))
4846, 15, 16, 19caovassd 5884 . . . . . . . . 9 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ((𝑎 +P 𝑓) +P 𝑑) = (𝑎 +P (𝑓 +P 𝑑)))
4947, 48eqtr4d 2150 . . . . . . . 8 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → (𝑓 +P (𝑎 +P 𝑑)) = ((𝑎 +P 𝑓) +P 𝑑))
5046, 16, 25, 22caovassd 5884 . . . . . . . . 9 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ((𝑓 +P 𝑏) +P 𝑐) = (𝑓 +P (𝑏 +P 𝑐)))
5116, 25, 22, 44, 46caov32d 5905 . . . . . . . . 9 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ((𝑓 +P 𝑏) +P 𝑐) = ((𝑓 +P 𝑐) +P 𝑏))
5250, 51eqtr3d 2149 . . . . . . . 8 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → (𝑓 +P (𝑏 +P 𝑐)) = ((𝑓 +P 𝑐) +P 𝑏))
5349, 52breq12d 3908 . . . . . . 7 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ((𝑓 +P (𝑎 +P 𝑑))<P (𝑓 +P (𝑏 +P 𝑐)) ↔ ((𝑎 +P 𝑓) +P 𝑑)<P ((𝑓 +P 𝑐) +P 𝑏)))
5442, 53bitrd 187 . . . . . 6 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ((𝑎 +P 𝑑)<P (𝑏 +P 𝑐) ↔ ((𝑎 +P 𝑓) +P 𝑑)<P ((𝑓 +P 𝑐) +P 𝑏)))
55 ltaprg 7375 . . . . . . . . 9 ((𝑟P𝑠P𝑡P) → (𝑟<P 𝑠 ↔ (𝑡 +P 𝑟)<P (𝑡 +P 𝑠)))
5655adantl 273 . . . . . . . 8 ((((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) ∧ (𝑟P𝑠P𝑡P)) → (𝑟<P 𝑠 ↔ (𝑡 +P 𝑟)<P (𝑡 +P 𝑠)))
5756, 18, 30, 19, 44caovord2d 5894 . . . . . . 7 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ((𝑎 +P 𝑓)<P (𝑏 +P 𝑒) ↔ ((𝑎 +P 𝑓) +P 𝑑)<P ((𝑏 +P 𝑒) +P 𝑑)))
58 addclpr 7293 . . . . . . . . . 10 ((𝑒P𝑑P) → (𝑒 +P 𝑑) ∈ P)
5928, 19, 58syl2anc 406 . . . . . . . . 9 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → (𝑒 +P 𝑑) ∈ P)
6056, 59, 24, 25, 44caovord2d 5894 . . . . . . . 8 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ((𝑒 +P 𝑑)<P (𝑓 +P 𝑐) ↔ ((𝑒 +P 𝑑) +P 𝑏)<P ((𝑓 +P 𝑐) +P 𝑏)))
6146, 25, 28, 19caovassd 5884 . . . . . . . . . 10 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ((𝑏 +P 𝑒) +P 𝑑) = (𝑏 +P (𝑒 +P 𝑑)))
6244, 25, 59caovcomd 5881 . . . . . . . . . 10 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → (𝑏 +P (𝑒 +P 𝑑)) = ((𝑒 +P 𝑑) +P 𝑏))
6361, 62eqtrd 2147 . . . . . . . . 9 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ((𝑏 +P 𝑒) +P 𝑑) = ((𝑒 +P 𝑑) +P 𝑏))
6463breq1d 3905 . . . . . . . 8 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → (((𝑏 +P 𝑒) +P 𝑑)<P ((𝑓 +P 𝑐) +P 𝑏) ↔ ((𝑒 +P 𝑑) +P 𝑏)<P ((𝑓 +P 𝑐) +P 𝑏)))
6560, 64bitr4d 190 . . . . . . 7 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ((𝑒 +P 𝑑)<P (𝑓 +P 𝑐) ↔ ((𝑏 +P 𝑒) +P 𝑑)<P ((𝑓 +P 𝑐) +P 𝑏)))
6657, 65orbi12d 765 . . . . . 6 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → (((𝑎 +P 𝑓)<P (𝑏 +P 𝑒) ∨ (𝑒 +P 𝑑)<P (𝑓 +P 𝑐)) ↔ (((𝑎 +P 𝑓) +P 𝑑)<P ((𝑏 +P 𝑒) +P 𝑑) ∨ ((𝑏 +P 𝑒) +P 𝑑)<P ((𝑓 +P 𝑐) +P 𝑏))))
6736, 54, 663imtr4d 202 . . . . 5 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ((𝑎 +P 𝑑)<P (𝑏 +P 𝑐) → ((𝑎 +P 𝑓)<P (𝑏 +P 𝑒) ∨ (𝑒 +P 𝑑)<P (𝑓 +P 𝑐))))
68 ltsrprg 7490 . . . . . 6 (((𝑎P𝑏P) ∧ (𝑐P𝑑P)) → ([⟨𝑎, 𝑏⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R ↔ (𝑎 +P 𝑑)<P (𝑏 +P 𝑐)))
69683adant3 984 . . . . 5 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ([⟨𝑎, 𝑏⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R ↔ (𝑎 +P 𝑑)<P (𝑏 +P 𝑐)))
70 ltsrprg 7490 . . . . . . 7 (((𝑎P𝑏P) ∧ (𝑒P𝑓P)) → ([⟨𝑎, 𝑏⟩] ~R <R [⟨𝑒, 𝑓⟩] ~R ↔ (𝑎 +P 𝑓)<P (𝑏 +P 𝑒)))
71703adant2 983 . . . . . 6 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ([⟨𝑎, 𝑏⟩] ~R <R [⟨𝑒, 𝑓⟩] ~R ↔ (𝑎 +P 𝑓)<P (𝑏 +P 𝑒)))
72 ltsrprg 7490 . . . . . . . 8 (((𝑒P𝑓P) ∧ (𝑐P𝑑P)) → ([⟨𝑒, 𝑓⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R ↔ (𝑒 +P 𝑑)<P (𝑓 +P 𝑐)))
7372ancoms 266 . . . . . . 7 (((𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ([⟨𝑒, 𝑓⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R ↔ (𝑒 +P 𝑑)<P (𝑓 +P 𝑐)))
74733adant1 982 . . . . . 6 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ([⟨𝑒, 𝑓⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R ↔ (𝑒 +P 𝑑)<P (𝑓 +P 𝑐)))
7571, 74orbi12d 765 . . . . 5 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → (([⟨𝑎, 𝑏⟩] ~R <R [⟨𝑒, 𝑓⟩] ~R ∨ [⟨𝑒, 𝑓⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R ) ↔ ((𝑎 +P 𝑓)<P (𝑏 +P 𝑒) ∨ (𝑒 +P 𝑑)<P (𝑓 +P 𝑐))))
7667, 69, 753imtr4d 202 . . . 4 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ([⟨𝑎, 𝑏⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R → ([⟨𝑎, 𝑏⟩] ~R <R [⟨𝑒, 𝑓⟩] ~R ∨ [⟨𝑒, 𝑓⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R )))
772, 6, 10, 14, 763ecoptocl 6472 . . 3 ((𝑥R𝑦R𝑧R) → (𝑥 <R 𝑦 → (𝑥 <R 𝑧𝑧 <R 𝑦)))
7877rgen3 2493 . 2 𝑥R𝑦R𝑧R (𝑥 <R 𝑦 → (𝑥 <R 𝑧𝑧 <R 𝑦))
79 df-iso 4179 . 2 ( <R Or R ↔ ( <R Po R ∧ ∀𝑥R𝑦R𝑧R (𝑥 <R 𝑦 → (𝑥 <R 𝑧𝑧 <R 𝑦))))
801, 78, 79mpbir2an 909 1 <R Or R
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wo 680  w3a 945   = wceq 1314  wcel 1463  wral 2390  cop 3496   class class class wbr 3895   Po wpo 4176   Or wor 4177  (class class class)co 5728  [cec 6381  Pcnp 7047   +P cpp 7049  <P cltp 7051   ~R cer 7052  Rcnr 7053   <R cltr 7059
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4003  ax-sep 4006  ax-nul 4014  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-iinf 4462
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-ral 2395  df-rex 2396  df-reu 2397  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-tr 3987  df-eprel 4171  df-id 4175  df-po 4178  df-iso 4179  df-iord 4248  df-on 4250  df-suc 4253  df-iom 4465  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-ov 5731  df-oprab 5732  df-mpo 5733  df-1st 5992  df-2nd 5993  df-recs 6156  df-irdg 6221  df-1o 6267  df-2o 6268  df-oadd 6271  df-omul 6272  df-er 6383  df-ec 6385  df-qs 6389  df-ni 7060  df-pli 7061  df-mi 7062  df-lti 7063  df-plpq 7100  df-mpq 7101  df-enq 7103  df-nqqs 7104  df-plqqs 7105  df-mqqs 7106  df-1nqqs 7107  df-rq 7108  df-ltnqqs 7109  df-enq0 7180  df-nq0 7181  df-0nq0 7182  df-plq0 7183  df-mq0 7184  df-inp 7222  df-iplp 7224  df-iltp 7226  df-enr 7469  df-nr 7470  df-ltr 7473
This theorem is referenced by:  1ne0sr  7509  addgt0sr  7518  caucvgsrlemcl  7531  caucvgsrlemfv  7533  axpre-ltirr  7617  axpre-ltwlin  7618  axpre-lttrn  7619
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