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Theorem noinfbnd1 27690
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 1191 . . . . . 6 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → 𝐵 No )
2 simpl 481 . . . . . . . 8 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
3 nominmo 27660 . . . . . . . . 9 (𝐵 No → ∃*𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
41, 3syl 17 . . . . . . . 8 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ∃*𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
5 reu5 3376 . . . . . . . 8 (∃!𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ↔ (∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ ∃*𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))
62, 4, 5sylanbrc 581 . . . . . . 7 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ∃!𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
7 riotacl 7400 . . . . . . 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 3983 . . . . 5 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ No )
10 noextendlt 27630 . . . . 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 1193 . . . . . 6 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → 𝑈𝐵)
13 nfv 1909 . . . . . . . . 9 𝑥(𝐵 No 𝐵𝑉𝑈𝐵)
14 nfcv 2899 . . . . . . . . . 10 𝑥𝐵
15 nfcv 2899 . . . . . . . . . . . 12 𝑥𝑦
16 nfcv 2899 . . . . . . . . . . . 12 𝑥 <s
17 nfriota1 7389 . . . . . . . . . . . 12 𝑥(𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
1815, 16, 17nfbr 5199 . . . . . . . . . . 11 𝑥 𝑦 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
1918nfn 1852 . . . . . . . . . 10 𝑥 ¬ 𝑦 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
2014, 19nfralw 3306 . . . . . . . . 9 𝑥𝑦𝐵 ¬ 𝑦 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
2113, 20nfim 1891 . . . . . . . 8 𝑥((𝐵 No 𝐵𝑉𝑈𝐵) → ∀𝑦𝐵 ¬ 𝑦 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))
22 simpl 481 . . . . . . . . . . 11 (((𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦 <s 𝑥) ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → (𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦 <s 𝑥))
23 rspe 3244 . . . . . . . . . . . . . 14 ((𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦 <s 𝑥) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
2423adantr 479 . . . . . . . . . . . . 13 (((𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦 <s 𝑥) ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
25 simpr1 1191 . . . . . . . . . . . . . 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 581 . . . . . . . . . . . 12 (((𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦 <s 𝑥) ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ∃!𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
28 riota1 7404 . . . . . . . . . . . 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 231 . . . . . . . . . 10 (((𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦 <s 𝑥) ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) = 𝑥)
31 simplr 767 . . . . . . . . . 10 (((𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦 <s 𝑥) ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ∀𝑦𝐵 ¬ 𝑦 <s 𝑥)
32 nfra1 3279 . . . . . . . . . . . . . 14 𝑦𝑦𝐵 ¬ 𝑦 <s 𝑥
33 nfcv 2899 . . . . . . . . . . . . . 14 𝑦𝐵
3432, 33nfriota 7395 . . . . . . . . . . . . 13 𝑦(𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
3534nfeq1 2915 . . . . . . . . . . . 12 𝑦(𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) = 𝑥
36 breq2 5156 . . . . . . . . . . . . 13 ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) = 𝑥 → (𝑦 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ↔ 𝑦 <s 𝑥))
3736notbid 317 . . . . . . . . . . . 12 ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) = 𝑥 → (¬ 𝑦 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ↔ ¬ 𝑦 <s 𝑥))
3835, 37ralbid 3268 . . . . . . . . . . 11 ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) = 𝑥 → (∀𝑦𝐵 ¬ 𝑦 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ↔ ∀𝑦𝐵 ¬ 𝑦 <s 𝑥))
3938biimprd 247 . . . . . . . . . 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 418 . . . . . . . 8 (𝑥𝐵 → (∀𝑦𝐵 ¬ 𝑦 <s 𝑥 → ((𝐵 No 𝐵𝑉𝑈𝐵) → ∀𝑦𝐵 ¬ 𝑦 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))))
4221, 41rexlimi 3254 . . . . . . 7 (∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 → ((𝐵 No 𝐵𝑉𝑈𝐵) → ∀𝑦𝐵 ¬ 𝑦 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)))
4342imp 405 . . . . . 6 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ∀𝑦𝐵 ¬ 𝑦 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))
44 nfcv 2899 . . . . . . . . 9 𝑦𝑈
45 nfcv 2899 . . . . . . . . 9 𝑦 <s
4644, 45, 34nfbr 5199 . . . . . . . 8 𝑦 𝑈 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
4746nfn 1852 . . . . . . 7 𝑦 ¬ 𝑈 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
48 breq1 5155 . . . . . . . 8 (𝑦 = 𝑈 → (𝑦 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ↔ 𝑈 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)))
4948notbid 317 . . . . . . 7 (𝑦 = 𝑈 → (¬ 𝑦 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ↔ ¬ 𝑈 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)))
5047, 49rspc 3599 . . . . . 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 27610 . . . . . . . . 9 ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ No → Fun (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))
53 funrel 6575 . . . . . . . . 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 6454 . . . . . . . 8 dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ⊆ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
56 relssres 6031 . . . . . . . 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 584 . . . . . . 7 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) = (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))
5857breq2d 5164 . . . . . 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 3983 . . . . . . 7 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → 𝑈 No )
60 nodmon 27611 . . . . . . . . 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 7828 . . . . . . . 8 (dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ On ↔ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ On)
6361, 62sylib 217 . . . . . . 7 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ On)
64 sltres 27623 . . . . . . 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 1368 . . . . . 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 259 . . . . 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 197 . . . 4 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ¬ (𝑈 ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))
68 1oex 8505 . . . . . . . 8 1o ∈ V
6968prid1 4771 . . . . . . 7 1o ∈ {1o, 2o}
7069noextend 27627 . . . . . 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 27621 . . . . . 6 ((𝑈 No ∧ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ On) → (𝑈 ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) ∈ No )
7359, 63, 72syl2anc 582 . . . . 5 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → (𝑈 ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) ∈ No )
74 sltso 27637 . . . . . 6 <s Or No
75 sotr3 5633 . . . . . 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 688 . . . . 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 1368 . . . 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 697 . . 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 4538 . . . . 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 2780 . . . 4 (∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥𝑇 = ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}))
8281adantr 479 . . 3 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → 𝑇 = ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}))
8381dmeqd 5912 . . . . . 6 (∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 → dom 𝑇 = dom ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}))
8468dmsnop 6225 . . . . . . . 8 dom {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩} = {dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)}
8584uneq2i 4161 . . . . . . 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 5917 . . . . . . 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 6380 . . . . . . 7 suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) = (dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)})
8885, 86, 873eqtr4i 2766 . . . . . 6 dom ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}) = suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
8983, 88eqtrdi 2784 . . . . 5 (∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 → dom 𝑇 = suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))
9089reseq2d 5989 . . . 4 (∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 → (𝑈 ↾ dom 𝑇) = (𝑈 ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)))
9190adantr 479 . . 3 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → (𝑈 ↾ dom 𝑇) = (𝑈 ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)))
9278, 82, 913brtr4d 5184 . 2 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → 𝑇 <s (𝑈 ↾ dom 𝑇))
93 simpl 481 . . 3 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
94 simpr1 1191 . . 3 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → 𝐵 No )
95 simpr2 1192 . . 3 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → 𝐵𝑉)
96 simpr3 1193 . . 3 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → 𝑈𝐵)
9779noinfbnd1lem6 27689 . . 3 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ 𝑈𝐵) → 𝑇 <s (𝑈 ↾ dom 𝑇))
9893, 94, 95, 96, 97syl121anc 1372 . 2 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → 𝑇 <s (𝑈 ↾ dom 𝑇))
9992, 98pm2.61ian 810 1 ((𝐵 No 𝐵𝑉𝑈𝐵) → 𝑇 <s (𝑈 ↾ dom 𝑇))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  {cab 2705  wral 3058  wrex 3067  ∃!wreu 3372  ∃*wrmo 3373  cun 3947  wss 3949  ifcif 4532  {csn 4632  cop 4638   class class class wbr 5152  cmpt 5235   Or wor 5593  dom cdm 5682  cres 5684  Rel wrel 5687  Oncon0 6374  suc csuc 6376  cio 6503  Fun wfun 6547  cfv 6553  crio 7381  1oc1o 8488  2oc2o 8489   No csur 27601   <s cslt 27602
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7748
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-tp 4637  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ord 6377  df-on 6378  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-1o 8495  df-2o 8496  df-no 27604  df-slt 27605  df-bday 27606
This theorem is referenced by:  noinfbnd2  27692  noetainflem3  27700
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