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Theorem nosupbnd1 31985
 Description: Bounding law from below for the surreal supremum. Proposition 4.2 of [Lipparini] p. 6. (Contributed by Scott Fenton, 6-Dec-2021.)
Hypothesis
Ref Expression
nosupbnd1.1 𝑆 = if(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2𝑜⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))
Assertion
Ref Expression
nosupbnd1 ((𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴) → (𝑈 ↾ dom 𝑆) <s 𝑆)
Distinct variable groups:   𝐴,𝑔,𝑢,𝑣,𝑥,𝑦   𝑢,𝑈,𝑣,𝑥,𝑦
Allowed substitution hints:   𝑆(𝑥,𝑦,𝑣,𝑢,𝑔)   𝑈(𝑔)

Proof of Theorem nosupbnd1
StepHypRef Expression
1 simpr3 1089 . . . . . 6 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → 𝑈𝐴)
2 nfv 1883 . . . . . . . . 9 𝑥(𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)
3 nfcv 2793 . . . . . . . . . 10 𝑥𝐴
4 nfriota1 6658 . . . . . . . . . . . 12 𝑥(𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
5 nfcv 2793 . . . . . . . . . . . 12 𝑥 <s
6 nfcv 2793 . . . . . . . . . . . 12 𝑥𝑦
74, 5, 6nfbr 4732 . . . . . . . . . . 11 𝑥(𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦
87nfn 1824 . . . . . . . . . 10 𝑥 ¬ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦
93, 8nfral 2974 . . . . . . . . 9 𝑥𝑦𝐴 ¬ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦
102, 9nfim 1865 . . . . . . . 8 𝑥((𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴) → ∀𝑦𝐴 ¬ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦)
11 simpl 472 . . . . . . . . . . 11 (((𝑥𝐴 ∧ ∀𝑦𝐴 ¬ 𝑥 <s 𝑦) ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → (𝑥𝐴 ∧ ∀𝑦𝐴 ¬ 𝑥 <s 𝑦))
12 rspe 3032 . . . . . . . . . . . . . 14 ((𝑥𝐴 ∧ ∀𝑦𝐴 ¬ 𝑥 <s 𝑦) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
1312adantr 480 . . . . . . . . . . . . 13 (((𝑥𝐴 ∧ ∀𝑦𝐴 ¬ 𝑥 <s 𝑦) ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
14 nomaxmo 31972 . . . . . . . . . . . . . . 15 (𝐴 No → ∃*𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
15143ad2ant1 1102 . . . . . . . . . . . . . 14 ((𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴) → ∃*𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
1615adantl 481 . . . . . . . . . . . . 13 (((𝑥𝐴 ∧ ∀𝑦𝐴 ¬ 𝑥 <s 𝑦) ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → ∃*𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
17 reu5 3189 . . . . . . . . . . . . 13 (∃!𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ↔ (∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ∃*𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦))
1813, 16, 17sylanbrc 699 . . . . . . . . . . . 12 (((𝑥𝐴 ∧ ∀𝑦𝐴 ¬ 𝑥 <s 𝑦) ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → ∃!𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
19 riota1 6669 . . . . . . . . . . . 12 (∃!𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 → ((𝑥𝐴 ∧ ∀𝑦𝐴 ¬ 𝑥 <s 𝑦) ↔ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) = 𝑥))
2018, 19syl 17 . . . . . . . . . . 11 (((𝑥𝐴 ∧ ∀𝑦𝐴 ¬ 𝑥 <s 𝑦) ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → ((𝑥𝐴 ∧ ∀𝑦𝐴 ¬ 𝑥 <s 𝑦) ↔ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) = 𝑥))
2111, 20mpbid 222 . . . . . . . . . 10 (((𝑥𝐴 ∧ ∀𝑦𝐴 ¬ 𝑥 <s 𝑦) ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) = 𝑥)
22 simplr 807 . . . . . . . . . 10 (((𝑥𝐴 ∧ ∀𝑦𝐴 ¬ 𝑥 <s 𝑦) ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → ∀𝑦𝐴 ¬ 𝑥 <s 𝑦)
23 nfra1 2970 . . . . . . . . . . . . . 14 𝑦𝑦𝐴 ¬ 𝑥 <s 𝑦
24 nfcv 2793 . . . . . . . . . . . . . 14 𝑦𝐴
2523, 24nfriota 6660 . . . . . . . . . . . . 13 𝑦(𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
26 nfcv 2793 . . . . . . . . . . . . 13 𝑦𝑥
2725, 26nfeq 2805 . . . . . . . . . . . 12 𝑦(𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) = 𝑥
28 breq1 4688 . . . . . . . . . . . . 13 ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) = 𝑥 → ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦𝑥 <s 𝑦))
2928notbid 307 . . . . . . . . . . . 12 ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) = 𝑥 → (¬ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦 ↔ ¬ 𝑥 <s 𝑦))
3027, 29ralbid 3012 . . . . . . . . . . 11 ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) = 𝑥 → (∀𝑦𝐴 ¬ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦 ↔ ∀𝑦𝐴 ¬ 𝑥 <s 𝑦))
3130biimprd 238 . . . . . . . . . 10 ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) = 𝑥 → (∀𝑦𝐴 ¬ 𝑥 <s 𝑦 → ∀𝑦𝐴 ¬ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦))
3221, 22, 31sylc 65 . . . . . . . . 9 (((𝑥𝐴 ∧ ∀𝑦𝐴 ¬ 𝑥 <s 𝑦) ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → ∀𝑦𝐴 ¬ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦)
3332exp31 629 . . . . . . . 8 (𝑥𝐴 → (∀𝑦𝐴 ¬ 𝑥 <s 𝑦 → ((𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴) → ∀𝑦𝐴 ¬ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦)))
3410, 33rexlimi 3053 . . . . . . 7 (∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 → ((𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴) → ∀𝑦𝐴 ¬ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦))
3534imp 444 . . . . . 6 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → ∀𝑦𝐴 ¬ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦)
36 nfcv 2793 . . . . . . . . 9 𝑦 <s
37 nfcv 2793 . . . . . . . . 9 𝑦𝑈
3825, 36, 37nfbr 4732 . . . . . . . 8 𝑦(𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑈
3938nfn 1824 . . . . . . 7 𝑦 ¬ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑈
40 breq2 4689 . . . . . . . 8 (𝑦 = 𝑈 → ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦 ↔ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑈))
4140notbid 307 . . . . . . 7 (𝑦 = 𝑈 → (¬ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦 ↔ ¬ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑈))
4239, 41rspc 3334 . . . . . 6 (𝑈𝐴 → (∀𝑦𝐴 ¬ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦 → ¬ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑈))
431, 35, 42sylc 65 . . . . 5 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → ¬ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑈)
44 simpr1 1087 . . . . . . . . . 10 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → 𝐴 No )
45 simpl 472 . . . . . . . . . . . 12 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
4615adantl 481 . . . . . . . . . . . 12 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → ∃*𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
4745, 46, 17sylanbrc 699 . . . . . . . . . . 11 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → ∃!𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
48 riotacl 6665 . . . . . . . . . . 11 (∃!𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 → (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∈ 𝐴)
4947, 48syl 17 . . . . . . . . . 10 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∈ 𝐴)
5044, 49sseldd 3637 . . . . . . . . 9 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∈ No )
51 nofun 31927 . . . . . . . . 9 ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∈ No → Fun (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦))
52 funrel 5943 . . . . . . . . 9 (Fun (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) → Rel (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦))
5350, 51, 523syl 18 . . . . . . . 8 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → Rel (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦))
54 sssucid 5840 . . . . . . . 8 dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ⊆ suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
55 relssres 5472 . . . . . . . 8 ((Rel (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∧ dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ⊆ suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)) → ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ↾ suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)) = (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦))
5653, 54, 55sylancl 695 . . . . . . 7 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ↾ suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)) = (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦))
5756breq1d 4695 . . . . . 6 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → (((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ↾ suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)) <s (𝑈 ↾ suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)) ↔ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s (𝑈 ↾ suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦))))
5844, 1sseldd 3637 . . . . . . 7 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → 𝑈 No )
59 nodmon 31928 . . . . . . . . 9 ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∈ No → dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∈ On)
6050, 59syl 17 . . . . . . . 8 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∈ On)
61 sucelon 7059 . . . . . . . 8 (dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∈ On ↔ suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∈ On)
6260, 61sylib 208 . . . . . . 7 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∈ On)
63 sltres 31940 . . . . . . 7 (((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∈ No 𝑈 No ∧ suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∈ On) → (((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ↾ suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)) <s (𝑈 ↾ suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)) → (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑈))
6450, 58, 62, 63syl3anc 1366 . . . . . 6 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → (((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ↾ suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)) <s (𝑈 ↾ suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)) → (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑈))
6557, 64sylbird 250 . . . . 5 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s (𝑈 ↾ suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)) → (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑈))
6643, 65mtod 189 . . . 4 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → ¬ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s (𝑈 ↾ suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)))
67 noextendgt 31948 . . . . 5 ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∈ No → (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2𝑜⟩}))
6850, 67syl 17 . . . 4 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2𝑜⟩}))
69 noreson 31938 . . . . . 6 ((𝑈 No ∧ suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∈ On) → (𝑈 ↾ suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)) ∈ No )
7058, 62, 69syl2anc 694 . . . . 5 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → (𝑈 ↾ suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)) ∈ No )
71 2on 7613 . . . . . . . . 9 2𝑜 ∈ On
7271elexi 3244 . . . . . . . 8 2𝑜 ∈ V
7372prid2 4330 . . . . . . 7 2𝑜 ∈ {1𝑜, 2𝑜}
7473noextend 31944 . . . . . 6 ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∈ No → ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2𝑜⟩}) ∈ No )
7550, 74syl 17 . . . . 5 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2𝑜⟩}) ∈ No )
76 sltso 31952 . . . . . 6 <s Or No
77 sotr2 5093 . . . . . 6 (( <s Or No ∧ ((𝑈 ↾ suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)) ∈ No ∧ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∈ No ∧ ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2𝑜⟩}) ∈ No )) → ((¬ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s (𝑈 ↾ suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)) ∧ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2𝑜⟩})) → (𝑈 ↾ suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)) <s ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2𝑜⟩})))
7876, 77mpan 706 . . . . 5 (((𝑈 ↾ suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)) ∈ No ∧ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∈ No ∧ ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2𝑜⟩}) ∈ No ) → ((¬ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s (𝑈 ↾ suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)) ∧ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2𝑜⟩})) → (𝑈 ↾ suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)) <s ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2𝑜⟩})))
7970, 50, 75, 78syl3anc 1366 . . . 4 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → ((¬ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s (𝑈 ↾ suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)) ∧ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2𝑜⟩})) → (𝑈 ↾ suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)) <s ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2𝑜⟩})))
8066, 68, 79mp2and 715 . . 3 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → (𝑈 ↾ suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)) <s ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2𝑜⟩}))
81 nosupbnd1.1 . . . . . . . 8 𝑆 = if(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2𝑜⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))
82 iftrue 4125 . . . . . . . 8 (∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 → if(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2𝑜⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥)))) = ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2𝑜⟩}))
8381, 82syl5eq 2697 . . . . . . 7 (∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦𝑆 = ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2𝑜⟩}))
8483dmeqd 5358 . . . . . 6 (∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 → dom 𝑆 = dom ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2𝑜⟩}))
8572dmsnop 5645 . . . . . . . 8 dom {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2𝑜⟩} = {dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)}
8685uneq2i 3797 . . . . . . 7 (dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ dom {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2𝑜⟩}) = (dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)})
87 dmun 5363 . . . . . . 7 dom ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2𝑜⟩}) = (dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ dom {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2𝑜⟩})
88 df-suc 5767 . . . . . . 7 suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) = (dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)})
8986, 87, 883eqtr4i 2683 . . . . . 6 dom ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2𝑜⟩}) = suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
9084, 89syl6eq 2701 . . . . 5 (∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 → dom 𝑆 = suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦))
9190adantr 480 . . . 4 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → dom 𝑆 = suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦))
9291reseq2d 5428 . . 3 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → (𝑈 ↾ dom 𝑆) = (𝑈 ↾ suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)))
9383adantr 480 . . 3 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → 𝑆 = ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2𝑜⟩}))
9480, 92, 933brtr4d 4717 . 2 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → (𝑈 ↾ dom 𝑆) <s 𝑆)
95 simpl 472 . . 3 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → ¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
96 simpr1 1087 . . 3 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → 𝐴 No )
97 simpr2 1088 . . 3 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → 𝐴 ∈ V)
98 simpr3 1089 . . 3 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → 𝑈𝐴)
9981nosupbnd1lem6 31984 . . 3 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → (𝑈 ↾ dom 𝑆) <s 𝑆)
10095, 96, 97, 98, 99syl121anc 1371 . 2 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → (𝑈 ↾ dom 𝑆) <s 𝑆)
10194, 100pm2.61ian 848 1 ((𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴) → (𝑈 ↾ dom 𝑆) <s 𝑆)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030  {cab 2637  ∀wral 2941  ∃wrex 2942  ∃!wreu 2943  ∃*wrmo 2944  Vcvv 3231   ∪ cun 3605   ⊆ wss 3607  ifcif 4119  {csn 4210  ⟨cop 4216   class class class wbr 4685   ↦ cmpt 4762   Or wor 5063  dom cdm 5143   ↾ cres 5145  Rel wrel 5148  Oncon0 5761  suc csuc 5763  ℩cio 5887  Fun wfun 5920  ‘cfv 5926  ℩crio 6650  1𝑜c1o 7598  2𝑜c2o 7599   No csur 31918
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