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Theorem ltsosr 7003
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 7002 . 2 <R Po R
2 df-nr 6966 . . . 4 R = ((P × P) / ~R )
3 breq1 3796 . . . . 5 ([⟨𝑎, 𝑏⟩] ~R = 𝑥 → ([⟨𝑎, 𝑏⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R𝑥 <R [⟨𝑐, 𝑑⟩] ~R ))
4 breq1 3796 . . . . . 6 ([⟨𝑎, 𝑏⟩] ~R = 𝑥 → ([⟨𝑎, 𝑏⟩] ~R <R [⟨𝑒, 𝑓⟩] ~R𝑥 <R [⟨𝑒, 𝑓⟩] ~R ))
54orbi1d 738 . . . . 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 232 . . . 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 3797 . . . . 5 ([⟨𝑐, 𝑑⟩] ~R = 𝑦 → (𝑥 <R [⟨𝑐, 𝑑⟩] ~R𝑥 <R 𝑦))
8 breq2 3797 . . . . . 6 ([⟨𝑐, 𝑑⟩] ~R = 𝑦 → ([⟨𝑒, 𝑓⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R ↔ [⟨𝑒, 𝑓⟩] ~R <R 𝑦))
98orbi2d 737 . . . . 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 232 . . . 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 3797 . . . . . 6 ([⟨𝑒, 𝑓⟩] ~R = 𝑧 → (𝑥 <R [⟨𝑒, 𝑓⟩] ~R𝑥 <R 𝑧))
12 breq1 3796 . . . . . 6 ([⟨𝑒, 𝑓⟩] ~R = 𝑧 → ([⟨𝑒, 𝑓⟩] ~R <R 𝑦𝑧 <R 𝑦))
1311, 12orbi12d 740 . . . . 5 ([⟨𝑒, 𝑓⟩] ~R = 𝑧 → ((𝑥 <R [⟨𝑒, 𝑓⟩] ~R ∨ [⟨𝑒, 𝑓⟩] ~R <R 𝑦) ↔ (𝑥 <R 𝑧𝑧 <R 𝑦)))
1413imbi2d 228 . . . 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 963 . . . . . . . . 9 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → 𝑎P)
16 simp3r 968 . . . . . . . . 9 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → 𝑓P)
17 addclpr 6789 . . . . . . . . 9 ((𝑎P𝑓P) → (𝑎 +P 𝑓) ∈ P)
1815, 16, 17syl2anc 403 . . . . . . . 8 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → (𝑎 +P 𝑓) ∈ P)
19 simp2r 966 . . . . . . . 8 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → 𝑑P)
20 addclpr 6789 . . . . . . . 8 (((𝑎 +P 𝑓) ∈ P𝑑P) → ((𝑎 +P 𝑓) +P 𝑑) ∈ P)
2118, 19, 20syl2anc 403 . . . . . . 7 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ((𝑎 +P 𝑓) +P 𝑑) ∈ P)
22 simp2l 965 . . . . . . . . 9 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → 𝑐P)
23 addclpr 6789 . . . . . . . . 9 ((𝑓P𝑐P) → (𝑓 +P 𝑐) ∈ P)
2416, 22, 23syl2anc 403 . . . . . . . 8 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → (𝑓 +P 𝑐) ∈ P)
25 simp1r 964 . . . . . . . 8 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → 𝑏P)
26 addclpr 6789 . . . . . . . 8 (((𝑓 +P 𝑐) ∈ P𝑏P) → ((𝑓 +P 𝑐) +P 𝑏) ∈ P)
2724, 25, 26syl2anc 403 . . . . . . 7 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ((𝑓 +P 𝑐) +P 𝑏) ∈ P)
28 simp3l 967 . . . . . . . . 9 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → 𝑒P)
29 addclpr 6789 . . . . . . . . 9 ((𝑏P𝑒P) → (𝑏 +P 𝑒) ∈ P)
3025, 28, 29syl2anc 403 . . . . . . . 8 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → (𝑏 +P 𝑒) ∈ P)
31 addclpr 6789 . . . . . . . 8 (((𝑏 +P 𝑒) ∈ P𝑑P) → ((𝑏 +P 𝑒) +P 𝑑) ∈ P)
3230, 19, 31syl2anc 403 . . . . . . 7 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ((𝑏 +P 𝑒) +P 𝑑) ∈ P)
33 ltsopr 6848 . . . . . . . 8 <P Or P
34 sowlin 4083 . . . . . . . 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 415 . . . . . . 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 1170 . . . . . 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 6789 . . . . . . . . 9 ((𝑎P𝑑P) → (𝑎 +P 𝑑) ∈ P)
3815, 19, 37syl2anc 403 . . . . . . . 8 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → (𝑎 +P 𝑑) ∈ P)
39 addclpr 6789 . . . . . . . . 9 ((𝑏P𝑐P) → (𝑏 +P 𝑐) ∈ P)
4025, 22, 39syl2anc 403 . . . . . . . 8 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → (𝑏 +P 𝑐) ∈ P)
41 ltaprg 6871 . . . . . . . 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 1170 . . . . . . 7 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ((𝑎 +P 𝑑)<P (𝑏 +P 𝑐) ↔ (𝑓 +P (𝑎 +P 𝑑))<P (𝑓 +P (𝑏 +P 𝑐))))
43 addcomprg 6830 . . . . . . . . . . 11 ((𝑟P𝑠P) → (𝑟 +P 𝑠) = (𝑠 +P 𝑟))
4443adantl 271 . . . . . . . . . 10 ((((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) ∧ (𝑟P𝑠P)) → (𝑟 +P 𝑠) = (𝑠 +P 𝑟))
45 addassprg 6831 . . . . . . . . . . 11 ((𝑟P𝑠P𝑡P) → ((𝑟 +P 𝑠) +P 𝑡) = (𝑟 +P (𝑠 +P 𝑡)))
4645adantl 271 . . . . . . . . . 10 ((((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) ∧ (𝑟P𝑠P𝑡P)) → ((𝑟 +P 𝑠) +P 𝑡) = (𝑟 +P (𝑠 +P 𝑡)))
4716, 15, 19, 44, 46caov12d 5713 . . . . . . . . 9 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → (𝑓 +P (𝑎 +P 𝑑)) = (𝑎 +P (𝑓 +P 𝑑)))
4846, 15, 16, 19caovassd 5691 . . . . . . . . 9 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ((𝑎 +P 𝑓) +P 𝑑) = (𝑎 +P (𝑓 +P 𝑑)))
4947, 48eqtr4d 2117 . . . . . . . 8 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → (𝑓 +P (𝑎 +P 𝑑)) = ((𝑎 +P 𝑓) +P 𝑑))
5046, 16, 25, 22caovassd 5691 . . . . . . . . 9 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ((𝑓 +P 𝑏) +P 𝑐) = (𝑓 +P (𝑏 +P 𝑐)))
5116, 25, 22, 44, 46caov32d 5712 . . . . . . . . 9 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ((𝑓 +P 𝑏) +P 𝑐) = ((𝑓 +P 𝑐) +P 𝑏))
5250, 51eqtr3d 2116 . . . . . . . 8 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → (𝑓 +P (𝑏 +P 𝑐)) = ((𝑓 +P 𝑐) +P 𝑏))
5349, 52breq12d 3806 . . . . . . 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 186 . . . . . 6 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ((𝑎 +P 𝑑)<P (𝑏 +P 𝑐) ↔ ((𝑎 +P 𝑓) +P 𝑑)<P ((𝑓 +P 𝑐) +P 𝑏)))
55 ltaprg 6871 . . . . . . . . 9 ((𝑟P𝑠P𝑡P) → (𝑟<P 𝑠 ↔ (𝑡 +P 𝑟)<P (𝑡 +P 𝑠)))
5655adantl 271 . . . . . . . 8 ((((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) ∧ (𝑟P𝑠P𝑡P)) → (𝑟<P 𝑠 ↔ (𝑡 +P 𝑟)<P (𝑡 +P 𝑠)))
5756, 18, 30, 19, 44caovord2d 5701 . . . . . . 7 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ((𝑎 +P 𝑓)<P (𝑏 +P 𝑒) ↔ ((𝑎 +P 𝑓) +P 𝑑)<P ((𝑏 +P 𝑒) +P 𝑑)))
58 addclpr 6789 . . . . . . . . . 10 ((𝑒P𝑑P) → (𝑒 +P 𝑑) ∈ P)
5928, 19, 58syl2anc 403 . . . . . . . . 9 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → (𝑒 +P 𝑑) ∈ P)
6056, 59, 24, 25, 44caovord2d 5701 . . . . . . . 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 5691 . . . . . . . . . 10 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ((𝑏 +P 𝑒) +P 𝑑) = (𝑏 +P (𝑒 +P 𝑑)))
6244, 25, 59caovcomd 5688 . . . . . . . . . 10 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → (𝑏 +P (𝑒 +P 𝑑)) = ((𝑒 +P 𝑑) +P 𝑏))
6361, 62eqtrd 2114 . . . . . . . . 9 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ((𝑏 +P 𝑒) +P 𝑑) = ((𝑒 +P 𝑑) +P 𝑏))
6463breq1d 3803 . . . . . . . 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 189 . . . . . . 7 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ((𝑒 +P 𝑑)<P (𝑓 +P 𝑐) ↔ ((𝑏 +P 𝑒) +P 𝑑)<P ((𝑓 +P 𝑐) +P 𝑏)))
6657, 65orbi12d 740 . . . . . 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 201 . . . . 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 6986 . . . . . 6 (((𝑎P𝑏P) ∧ (𝑐P𝑑P)) → ([⟨𝑎, 𝑏⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R ↔ (𝑎 +P 𝑑)<P (𝑏 +P 𝑐)))
69683adant3 959 . . . . 5 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ([⟨𝑎, 𝑏⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R ↔ (𝑎 +P 𝑑)<P (𝑏 +P 𝑐)))
70 ltsrprg 6986 . . . . . . 7 (((𝑎P𝑏P) ∧ (𝑒P𝑓P)) → ([⟨𝑎, 𝑏⟩] ~R <R [⟨𝑒, 𝑓⟩] ~R ↔ (𝑎 +P 𝑓)<P (𝑏 +P 𝑒)))
71703adant2 958 . . . . . 6 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ([⟨𝑎, 𝑏⟩] ~R <R [⟨𝑒, 𝑓⟩] ~R ↔ (𝑎 +P 𝑓)<P (𝑏 +P 𝑒)))
72 ltsrprg 6986 . . . . . . . 8 (((𝑒P𝑓P) ∧ (𝑐P𝑑P)) → ([⟨𝑒, 𝑓⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R ↔ (𝑒 +P 𝑑)<P (𝑓 +P 𝑐)))
7372ancoms 264 . . . . . . 7 (((𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ([⟨𝑒, 𝑓⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R ↔ (𝑒 +P 𝑑)<P (𝑓 +P 𝑐)))
74733adant1 957 . . . . . 6 (((𝑎P𝑏P) ∧ (𝑐P𝑑P) ∧ (𝑒P𝑓P)) → ([⟨𝑒, 𝑓⟩] ~R <R [⟨𝑐, 𝑑⟩] ~R ↔ (𝑒 +P 𝑑)<P (𝑓 +P 𝑐)))
7571, 74orbi12d 740 . . . . 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 201 . . . 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 6261 . . 3 ((𝑥R𝑦R𝑧R) → (𝑥 <R 𝑦 → (𝑥 <R 𝑧𝑧 <R 𝑦)))
7877rgen3 2449 . 2 𝑥R𝑦R𝑧R (𝑥 <R 𝑦 → (𝑥 <R 𝑧𝑧 <R 𝑦))
79 df-iso 4060 . 2 ( <R Or R ↔ ( <R Po R ∧ ∀𝑥R𝑦R𝑧R (𝑥 <R 𝑦 → (𝑥 <R 𝑧𝑧 <R 𝑦))))
801, 78, 79mpbir2an 884 1 <R Or R
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wo 662  w3a 920   = wceq 1285  wcel 1434  wral 2349  cop 3409   class class class wbr 3793   Po wpo 4057   Or wor 4058  (class class class)co 5543  [cec 6170  Pcnp 6543   +P cpp 6545  <P cltp 6547   ~R cer 6548  Rcnr 6549   <R cltr 6555
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-eprel 4052  df-id 4056  df-po 4059  df-iso 4060  df-iord 4129  df-on 4131  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-irdg 6019  df-1o 6065  df-2o 6066  df-oadd 6069  df-omul 6070  df-er 6172  df-ec 6174  df-qs 6178  df-ni 6556  df-pli 6557  df-mi 6558  df-lti 6559  df-plpq 6596  df-mpq 6597  df-enq 6599  df-nqqs 6600  df-plqqs 6601  df-mqqs 6602  df-1nqqs 6603  df-rq 6604  df-ltnqqs 6605  df-enq0 6676  df-nq0 6677  df-0nq0 6678  df-plq0 6679  df-mq0 6680  df-inp 6718  df-iplp 6720  df-iltp 6722  df-enr 6965  df-nr 6966  df-ltr 6969
This theorem is referenced by:  1ne0sr  7005  addgt0sr  7014  caucvgsrlemcl  7027  caucvgsrlemfv  7029  axpre-ltirr  7110  axpre-ltwlin  7111  axpre-lttrn  7112
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