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Theorem noinfbnd1 27789
Description: Bounding law from above for the surreal infimum. Analagous to proposition 4.2 of [Lipparini] p. 6. (Contributed by Scott Fenton, 9-Aug-2024.)
Hypothesis
Ref Expression
noinfbnd1.1 𝑇 = if(∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥, ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))
Assertion
Ref Expression
noinfbnd1 ((𝐵 No 𝐵𝑉𝑈𝐵) → 𝑇 <s (𝑈 ↾ dom 𝑇))
Distinct variable groups:   𝐵,𝑔,𝑢,𝑣,𝑥,𝑦   𝑣,𝑈   𝑥,𝑢,𝑦   𝑔,𝑉   𝑥,𝑣,𝑦,𝑈   𝑥,𝑉
Allowed substitution hints:   𝑇(𝑥,𝑦,𝑣,𝑢,𝑔)   𝑈(𝑢,𝑔)   𝑉(𝑦,𝑣,𝑢)

Proof of Theorem noinfbnd1
StepHypRef Expression
1 simpr1 1193 . . . . . 6 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → 𝐵 No )
2 simpl 482 . . . . . . . 8 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
3 nominmo 27759 . . . . . . . . 9 (𝐵 No → ∃*𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
41, 3syl 17 . . . . . . . 8 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ∃*𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
5 reu5 3380 . . . . . . . 8 (∃!𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ↔ (∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ ∃*𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))
62, 4, 5sylanbrc 583 . . . . . . 7 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ∃!𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
7 riotacl 7405 . . . . . . 7 (∃!𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 → (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ 𝐵)
86, 7syl 17 . . . . . 6 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ 𝐵)
91, 8sseldd 3996 . . . . 5 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ No )
10 noextendlt 27729 . . . . 5 ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ No → ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}) <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))
119, 10syl 17 . . . 4 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}) <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))
12 simpr3 1195 . . . . . 6 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → 𝑈𝐵)
13 nfv 1912 . . . . . . . . 9 𝑥(𝐵 No 𝐵𝑉𝑈𝐵)
14 nfcv 2903 . . . . . . . . . 10 𝑥𝐵
15 nfcv 2903 . . . . . . . . . . . 12 𝑥𝑦
16 nfcv 2903 . . . . . . . . . . . 12 𝑥 <s
17 nfriota1 7395 . . . . . . . . . . . 12 𝑥(𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
1815, 16, 17nfbr 5195 . . . . . . . . . . 11 𝑥 𝑦 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
1918nfn 1855 . . . . . . . . . 10 𝑥 ¬ 𝑦 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
2014, 19nfralw 3309 . . . . . . . . 9 𝑥𝑦𝐵 ¬ 𝑦 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
2113, 20nfim 1894 . . . . . . . 8 𝑥((𝐵 No 𝐵𝑉𝑈𝐵) → ∀𝑦𝐵 ¬ 𝑦 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))
22 simpl 482 . . . . . . . . . . 11 (((𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦 <s 𝑥) ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → (𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦 <s 𝑥))
23 rspe 3247 . . . . . . . . . . . . . 14 ((𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦 <s 𝑥) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
2423adantr 480 . . . . . . . . . . . . 13 (((𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦 <s 𝑥) ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
25 simpr1 1193 . . . . . . . . . . . . . 14 (((𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦 <s 𝑥) ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → 𝐵 No )
2625, 3syl 17 . . . . . . . . . . . . 13 (((𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦 <s 𝑥) ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ∃*𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
2724, 26, 5sylanbrc 583 . . . . . . . . . . . 12 (((𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦 <s 𝑥) ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ∃!𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
28 riota1 7409 . . . . . . . . . . . 12 (∃!𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 → ((𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦 <s 𝑥) ↔ (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) = 𝑥))
2927, 28syl 17 . . . . . . . . . . 11 (((𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦 <s 𝑥) ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ((𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦 <s 𝑥) ↔ (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) = 𝑥))
3022, 29mpbid 232 . . . . . . . . . 10 (((𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦 <s 𝑥) ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) = 𝑥)
31 simplr 769 . . . . . . . . . 10 (((𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦 <s 𝑥) ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ∀𝑦𝐵 ¬ 𝑦 <s 𝑥)
32 nfra1 3282 . . . . . . . . . . . . . 14 𝑦𝑦𝐵 ¬ 𝑦 <s 𝑥
33 nfcv 2903 . . . . . . . . . . . . . 14 𝑦𝐵
3432, 33nfriota 7400 . . . . . . . . . . . . 13 𝑦(𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
3534nfeq1 2919 . . . . . . . . . . . 12 𝑦(𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) = 𝑥
36 breq2 5152 . . . . . . . . . . . . 13 ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) = 𝑥 → (𝑦 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ↔ 𝑦 <s 𝑥))
3736notbid 318 . . . . . . . . . . . 12 ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) = 𝑥 → (¬ 𝑦 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ↔ ¬ 𝑦 <s 𝑥))
3835, 37ralbid 3271 . . . . . . . . . . 11 ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) = 𝑥 → (∀𝑦𝐵 ¬ 𝑦 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ↔ ∀𝑦𝐵 ¬ 𝑦 <s 𝑥))
3938biimprd 248 . . . . . . . . . 10 ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) = 𝑥 → (∀𝑦𝐵 ¬ 𝑦 <s 𝑥 → ∀𝑦𝐵 ¬ 𝑦 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)))
4030, 31, 39sylc 65 . . . . . . . . 9 (((𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦 <s 𝑥) ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ∀𝑦𝐵 ¬ 𝑦 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))
4140exp31 419 . . . . . . . 8 (𝑥𝐵 → (∀𝑦𝐵 ¬ 𝑦 <s 𝑥 → ((𝐵 No 𝐵𝑉𝑈𝐵) → ∀𝑦𝐵 ¬ 𝑦 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))))
4221, 41rexlimi 3257 . . . . . . 7 (∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 → ((𝐵 No 𝐵𝑉𝑈𝐵) → ∀𝑦𝐵 ¬ 𝑦 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)))
4342imp 406 . . . . . 6 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ∀𝑦𝐵 ¬ 𝑦 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))
44 nfcv 2903 . . . . . . . . 9 𝑦𝑈
45 nfcv 2903 . . . . . . . . 9 𝑦 <s
4644, 45, 34nfbr 5195 . . . . . . . 8 𝑦 𝑈 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
4746nfn 1855 . . . . . . 7 𝑦 ¬ 𝑈 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
48 breq1 5151 . . . . . . . 8 (𝑦 = 𝑈 → (𝑦 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ↔ 𝑈 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)))
4948notbid 318 . . . . . . 7 (𝑦 = 𝑈 → (¬ 𝑦 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ↔ ¬ 𝑈 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)))
5047, 49rspc 3610 . . . . . 6 (𝑈𝐵 → (∀𝑦𝐵 ¬ 𝑦 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) → ¬ 𝑈 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)))
5112, 43, 50sylc 65 . . . . 5 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ¬ 𝑈 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))
52 nofun 27709 . . . . . . . . 9 ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ No → Fun (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))
53 funrel 6585 . . . . . . . . 9 (Fun (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) → Rel (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))
549, 52, 533syl 18 . . . . . . . 8 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → Rel (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))
55 sssucid 6466 . . . . . . . 8 dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ⊆ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
56 relssres 6042 . . . . . . . 8 ((Rel (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∧ dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ⊆ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) → ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) = (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))
5754, 55, 56sylancl 586 . . . . . . 7 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) = (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))
5857breq2d 5160 . . . . . 6 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ((𝑈 ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) <s ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) ↔ (𝑈 ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)))
591, 12sseldd 3996 . . . . . . 7 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → 𝑈 No )
60 nodmon 27710 . . . . . . . . 9 ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ No → dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ On)
619, 60syl 17 . . . . . . . 8 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ On)
62 onsucb 7837 . . . . . . . 8 (dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ On ↔ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ On)
6361, 62sylib 218 . . . . . . 7 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ On)
64 sltres 27722 . . . . . . 7 ((𝑈 No ∧ (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ No ∧ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ On) → ((𝑈 ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) <s ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) → 𝑈 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)))
6559, 9, 63, 64syl3anc 1370 . . . . . 6 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ((𝑈 ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) <s ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) → 𝑈 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)))
6658, 65sylbird 260 . . . . 5 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ((𝑈 ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) → 𝑈 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)))
6751, 66mtod 198 . . . 4 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ¬ (𝑈 ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))
68 1oex 8515 . . . . . . . 8 1o ∈ V
6968prid1 4767 . . . . . . 7 1o ∈ {1o, 2o}
7069noextend 27726 . . . . . 6 ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ No → ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}) ∈ No )
719, 70syl 17 . . . . 5 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}) ∈ No )
72 noreson 27720 . . . . . 6 ((𝑈 No ∧ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ On) → (𝑈 ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) ∈ No )
7359, 63, 72syl2anc 584 . . . . 5 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → (𝑈 ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) ∈ No )
74 sltso 27736 . . . . . 6 <s Or No
75 sotr3 5637 . . . . . 6 (( <s Or No ∧ (((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}) ∈ No ∧ (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ No ∧ (𝑈 ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) ∈ No )) → ((((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}) <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∧ ¬ (𝑈 ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) → ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}) <s (𝑈 ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))))
7674, 75mpan 690 . . . . 5 ((((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}) ∈ No ∧ (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ No ∧ (𝑈 ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) ∈ No ) → ((((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}) <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∧ ¬ (𝑈 ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) → ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}) <s (𝑈 ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))))
7771, 9, 73, 76syl3anc 1370 . . . 4 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ((((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}) <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∧ ¬ (𝑈 ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) → ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}) <s (𝑈 ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))))
7811, 67, 77mp2and 699 . . 3 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}) <s (𝑈 ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)))
79 noinfbnd1.1 . . . . 5 𝑇 = if(∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥, ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))
80 iftrue 4537 . . . . 5 (∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 → if(∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥, ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)))) = ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}))
8179, 80eqtrid 2787 . . . 4 (∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥𝑇 = ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}))
8281adantr 480 . . 3 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → 𝑇 = ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}))
8381dmeqd 5919 . . . . . 6 (∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 → dom 𝑇 = dom ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}))
8468dmsnop 6238 . . . . . . . 8 dom {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩} = {dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)}
8584uneq2i 4175 . . . . . . 7 (dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ dom {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}) = (dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)})
86 dmun 5924 . . . . . . 7 dom ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}) = (dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ dom {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩})
87 df-suc 6392 . . . . . . 7 suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) = (dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)})
8885, 86, 873eqtr4i 2773 . . . . . 6 dom ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}) = suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
8983, 88eqtrdi 2791 . . . . 5 (∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 → dom 𝑇 = suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))
9089reseq2d 6000 . . . 4 (∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 → (𝑈 ↾ dom 𝑇) = (𝑈 ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)))
9190adantr 480 . . 3 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → (𝑈 ↾ dom 𝑇) = (𝑈 ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)))
9278, 82, 913brtr4d 5180 . 2 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → 𝑇 <s (𝑈 ↾ dom 𝑇))
93 simpl 482 . . 3 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
94 simpr1 1193 . . 3 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → 𝐵 No )
95 simpr2 1194 . . 3 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → 𝐵𝑉)
96 simpr3 1195 . . 3 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → 𝑈𝐵)
9779noinfbnd1lem6 27788 . . 3 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ 𝑈𝐵) → 𝑇 <s (𝑈 ↾ dom 𝑇))
9893, 94, 95, 96, 97syl121anc 1374 . 2 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → 𝑇 <s (𝑈 ↾ dom 𝑇))
9992, 98pm2.61ian 812 1 ((𝐵 No 𝐵𝑉𝑈𝐵) → 𝑇 <s (𝑈 ↾ dom 𝑇))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  {cab 2712  wral 3059  wrex 3068  ∃!wreu 3376  ∃*wrmo 3377  cun 3961  wss 3963  ifcif 4531  {csn 4631  cop 4637   class class class wbr 5148  cmpt 5231   Or wor 5596  dom cdm 5689  cres 5691  Rel wrel 5694  Oncon0 6386  suc csuc 6388  cio 6514  Fun wfun 6557  cfv 6563  crio 7387  1oc1o 8498  2oc2o 8499   No csur 27699   <s cslt 27700
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571  df-riota 7388  df-1o 8505  df-2o 8506  df-no 27702  df-slt 27703  df-bday 27704
This theorem is referenced by:  noinfbnd2  27791  noetainflem3  27799
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