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Theorem noinfbnd1 33497
 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 486 . . . . . . . 8 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
3 nominmo 33467 . . . . . . . . 9 (𝐵 No → ∃*𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
41, 3syl 17 . . . . . . . 8 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ∃*𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
5 reu5 3340 . . . . . . . 8 (∃!𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ↔ (∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ ∃*𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))
62, 4, 5sylanbrc 586 . . . . . . 7 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ∃!𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
7 riotacl 7125 . . . . . . 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 3893 . . . . 5 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ No )
10 noextendlt 33437 . . . . 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 1915 . . . . . . . . 9 𝑥(𝐵 No 𝐵𝑉𝑈𝐵)
14 nfcv 2919 . . . . . . . . . 10 𝑥𝐵
15 nfcv 2919 . . . . . . . . . . . 12 𝑥𝑦
16 nfcv 2919 . . . . . . . . . . . 12 𝑥 <s
17 nfriota1 7115 . . . . . . . . . . . 12 𝑥(𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
1815, 16, 17nfbr 5079 . . . . . . . . . . 11 𝑥 𝑦 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
1918nfn 1858 . . . . . . . . . 10 𝑥 ¬ 𝑦 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
2014, 19nfralw 3153 . . . . . . . . 9 𝑥𝑦𝐵 ¬ 𝑦 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
2113, 20nfim 1897 . . . . . . . 8 𝑥((𝐵 No 𝐵𝑉𝑈𝐵) → ∀𝑦𝐵 ¬ 𝑦 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))
22 simpl 486 . . . . . . . . . . 11 (((𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦 <s 𝑥) ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → (𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦 <s 𝑥))
23 rspe 3228 . . . . . . . . . . . . . 14 ((𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦 <s 𝑥) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
2423adantr 484 . . . . . . . . . . . . 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 586 . . . . . . . . . . . 12 (((𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦 <s 𝑥) ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ∃!𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
28 riota1 7129 . . . . . . . . . . . 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 235 . . . . . . . . . 10 (((𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦 <s 𝑥) ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) = 𝑥)
31 simplr 768 . . . . . . . . . 10 (((𝑥𝐵 ∧ ∀𝑦𝐵 ¬ 𝑦 <s 𝑥) ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ∀𝑦𝐵 ¬ 𝑦 <s 𝑥)
32 nfra1 3147 . . . . . . . . . . . . . 14 𝑦𝑦𝐵 ¬ 𝑦 <s 𝑥
33 nfcv 2919 . . . . . . . . . . . . . 14 𝑦𝐵
3432, 33nfriota 7120 . . . . . . . . . . . . 13 𝑦(𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
3534nfeq1 2934 . . . . . . . . . . . 12 𝑦(𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) = 𝑥
36 breq2 5036 . . . . . . . . . . . . 13 ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) = 𝑥 → (𝑦 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ↔ 𝑦 <s 𝑥))
3736notbid 321 . . . . . . . . . . . 12 ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) = 𝑥 → (¬ 𝑦 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ↔ ¬ 𝑦 <s 𝑥))
3835, 37ralbid 3159 . . . . . . . . . . 11 ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) = 𝑥 → (∀𝑦𝐵 ¬ 𝑦 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ↔ ∀𝑦𝐵 ¬ 𝑦 <s 𝑥))
3938biimprd 251 . . . . . . . . . 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 423 . . . . . . . 8 (𝑥𝐵 → (∀𝑦𝐵 ¬ 𝑦 <s 𝑥 → ((𝐵 No 𝐵𝑉𝑈𝐵) → ∀𝑦𝐵 ¬ 𝑦 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))))
4221, 41rexlimi 3239 . . . . . . 7 (∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 → ((𝐵 No 𝐵𝑉𝑈𝐵) → ∀𝑦𝐵 ¬ 𝑦 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)))
4342imp 410 . . . . . 6 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ∀𝑦𝐵 ¬ 𝑦 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))
44 nfcv 2919 . . . . . . . . 9 𝑦𝑈
45 nfcv 2919 . . . . . . . . 9 𝑦 <s
4644, 45, 34nfbr 5079 . . . . . . . 8 𝑦 𝑈 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
4746nfn 1858 . . . . . . 7 𝑦 ¬ 𝑈 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
48 breq1 5035 . . . . . . . 8 (𝑦 = 𝑈 → (𝑦 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ↔ 𝑈 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)))
4948notbid 321 . . . . . . 7 (𝑦 = 𝑈 → (¬ 𝑦 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ↔ ¬ 𝑈 <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)))
5047, 49rspc 3529 . . . . . 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 33417 . . . . . . . . 9 ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ No → Fun (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))
53 funrel 6352 . . . . . . . . 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 6246 . . . . . . . 8 dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ⊆ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
56 relssres 5864 . . . . . . . 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 589 . . . . . . 7 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) = (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))
5857breq2d 5044 . . . . . 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 3893 . . . . . . 7 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → 𝑈 No )
60 nodmon 33418 . . . . . . . . 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 sucelon 7531 . . . . . . . 8 (dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ On ↔ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ On)
6361, 62sylib 221 . . . . . . 7 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ On)
64 sltres 33430 . . . . . . 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 263 . . . . 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 201 . . . 4 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → ¬ (𝑈 ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) <s (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))
68 1oex 8120 . . . . . . . 8 1o ∈ V
6968prid1 4655 . . . . . . 7 1o ∈ {1o, 2o}
7069noextend 33434 . . . . . 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 33428 . . . . . 6 ((𝑈 No ∧ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∈ On) → (𝑈 ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) ∈ No )
7359, 63, 72syl2anc 587 . . . . 5 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → (𝑈 ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)) ∈ No )
74 sltso 33444 . . . . . 6 <s Or No
75 sotr3 33249 . . . . . 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 689 . . . . 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 698 . . 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 4426 . . . . 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, 80syl5eq 2805 . . . 4 (∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥𝑇 = ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}))
8281adantr 484 . . 3 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → 𝑇 = ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}))
8381dmeqd 5745 . . . . . 6 (∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 → dom 𝑇 = dom ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}))
8468dmsnop 6045 . . . . . . . 8 dom {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩} = {dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)}
8584uneq2i 4065 . . . . . . 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 5750 . . . . . . 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 6175 . . . . . . 7 suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) = (dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)})
8885, 86, 873eqtr4i 2791 . . . . . 6 dom ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}) = suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
8983, 88eqtrdi 2809 . . . . 5 (∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 → dom 𝑇 = suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥))
9089reseq2d 5823 . . . 4 (∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 → (𝑈 ↾ dom 𝑇) = (𝑈 ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)))
9190adantr 484 . . 3 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → (𝑈 ↾ dom 𝑇) = (𝑈 ↾ suc dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)))
9278, 82, 913brtr4d 5064 . 2 ((∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉𝑈𝐵)) → 𝑇 <s (𝑈 ↾ dom 𝑇))
93 simpl 486 . . 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 33496 . . 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 811 1 ((𝐵 No 𝐵𝑉𝑈𝐵) → 𝑇 <s (𝑈 ↾ dom 𝑇))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  {cab 2735  ∀wral 3070  ∃wrex 3071  ∃!wreu 3072  ∃*wrmo 3073   ∪ cun 3856   ⊆ wss 3858  ifcif 4420  {csn 4522  ⟨cop 4528   class class class wbr 5032   ↦ cmpt 5112   Or wor 5442  dom cdm 5524   ↾ cres 5526  Rel wrel 5529  Oncon0 6169  suc csuc 6171  ℩cio 6292  Fun wfun 6329  ‘cfv 6335  ℩crio 7107  1oc1o 8105  2oc2o 8106   No csur 33408
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