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Theorem nosupbnd1 33328
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 1193 . . . . . 6 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → 𝑈𝐴)
2 nfv 1915 . . . . . . . . 9 𝑥(𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)
3 nfcv 2958 . . . . . . . . . 10 𝑥𝐴
4 nfriota1 7104 . . . . . . . . . . . 12 𝑥(𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
5 nfcv 2958 . . . . . . . . . . . 12 𝑥 <s
6 nfcv 2958 . . . . . . . . . . . 12 𝑥𝑦
74, 5, 6nfbr 5080 . . . . . . . . . . 11 𝑥(𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦
87nfn 1858 . . . . . . . . . 10 𝑥 ¬ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦
93, 8nfralw 3192 . . . . . . . . 9 𝑥𝑦𝐴 ¬ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦
102, 9nfim 1897 . . . . . . . 8 𝑥((𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴) → ∀𝑦𝐴 ¬ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦)
11 simpl 486 . . . . . . . . . . 11 (((𝑥𝐴 ∧ ∀𝑦𝐴 ¬ 𝑥 <s 𝑦) ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → (𝑥𝐴 ∧ ∀𝑦𝐴 ¬ 𝑥 <s 𝑦))
12 rspe 3266 . . . . . . . . . . . . . 14 ((𝑥𝐴 ∧ ∀𝑦𝐴 ¬ 𝑥 <s 𝑦) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
1312adantr 484 . . . . . . . . . . . . 13 (((𝑥𝐴 ∧ ∀𝑦𝐴 ¬ 𝑥 <s 𝑦) ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
14 nomaxmo 33315 . . . . . . . . . . . . . . 15 (𝐴 No → ∃*𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
15143ad2ant1 1130 . . . . . . . . . . . . . 14 ((𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴) → ∃*𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
1615adantl 485 . . . . . . . . . . . . 13 (((𝑥𝐴 ∧ ∀𝑦𝐴 ¬ 𝑥 <s 𝑦) ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → ∃*𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
17 reu5 3378 . . . . . . . . . . . . 13 (∃!𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ↔ (∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ∃*𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦))
1813, 16, 17sylanbrc 586 . . . . . . . . . . . 12 (((𝑥𝐴 ∧ ∀𝑦𝐴 ¬ 𝑥 <s 𝑦) ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → ∃!𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
19 riota1 7118 . . . . . . . . . . . 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 235 . . . . . . . . . 10 (((𝑥𝐴 ∧ ∀𝑦𝐴 ¬ 𝑥 <s 𝑦) ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) = 𝑥)
22 simplr 768 . . . . . . . . . 10 (((𝑥𝐴 ∧ ∀𝑦𝐴 ¬ 𝑥 <s 𝑦) ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → ∀𝑦𝐴 ¬ 𝑥 <s 𝑦)
23 nfra1 3186 . . . . . . . . . . . . . 14 𝑦𝑦𝐴 ¬ 𝑥 <s 𝑦
24 nfcv 2958 . . . . . . . . . . . . . 14 𝑦𝐴
2523, 24nfriota 7109 . . . . . . . . . . . . 13 𝑦(𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
26 nfcv 2958 . . . . . . . . . . . . 13 𝑦𝑥
2725, 26nfeq 2971 . . . . . . . . . . . 12 𝑦(𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) = 𝑥
28 breq1 5036 . . . . . . . . . . . . 13 ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) = 𝑥 → ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦𝑥 <s 𝑦))
2928notbid 321 . . . . . . . . . . . 12 ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) = 𝑥 → (¬ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦 ↔ ¬ 𝑥 <s 𝑦))
3027, 29ralbid 3198 . . . . . . . . . . 11 ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) = 𝑥 → (∀𝑦𝐴 ¬ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦 ↔ ∀𝑦𝐴 ¬ 𝑥 <s 𝑦))
3130biimprd 251 . . . . . . . . . 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 423 . . . . . . . 8 (𝑥𝐴 → (∀𝑦𝐴 ¬ 𝑥 <s 𝑦 → ((𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴) → ∀𝑦𝐴 ¬ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦)))
3410, 33rexlimi 3277 . . . . . . 7 (∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 → ((𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴) → ∀𝑦𝐴 ¬ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦))
3534imp 410 . . . . . 6 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → ∀𝑦𝐴 ¬ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦)
36 nfcv 2958 . . . . . . . . 9 𝑦 <s
37 nfcv 2958 . . . . . . . . 9 𝑦𝑈
3825, 36, 37nfbr 5080 . . . . . . . 8 𝑦(𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑈
3938nfn 1858 . . . . . . 7 𝑦 ¬ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑈
40 breq2 5037 . . . . . . . 8 (𝑦 = 𝑈 → ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦 ↔ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑈))
4140notbid 321 . . . . . . 7 (𝑦 = 𝑈 → (¬ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦 ↔ ¬ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑈))
4239, 41rspc 3562 . . . . . 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 1191 . . . . . . . . . 10 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → 𝐴 No )
45 simpl 486 . . . . . . . . . . . 12 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
4615adantl 485 . . . . . . . . . . . 12 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → ∃*𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
4745, 46, 17sylanbrc 586 . . . . . . . . . . 11 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → ∃!𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
48 riotacl 7114 . . . . . . . . . . 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 3919 . . . . . . . . 9 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∈ No )
51 nofun 33270 . . . . . . . . 9 ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∈ No → Fun (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦))
52 funrel 6345 . . . . . . . . 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 6240 . . . . . . . 8 dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ⊆ suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
55 relssres 5863 . . . . . . . 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 589 . . . . . . 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 5043 . . . . . 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 3919 . . . . . . 7 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → 𝑈 No )
59 nodmon 33271 . . . . . . . . 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 7516 . . . . . . . 8 (dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∈ On ↔ suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∈ On)
6260, 61sylib 221 . . . . . . 7 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∈ On)
63 sltres 33283 . . . . . . 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 1368 . . . . . 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 263 . . . . 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 201 . . . 4 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → ¬ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s (𝑈 ↾ suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)))
67 noextendgt 33291 . . . . 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 33281 . . . . . 6 ((𝑈 No ∧ suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∈ On) → (𝑈 ↾ suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)) ∈ No )
7058, 62, 69syl2anc 587 . . . . 5 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → (𝑈 ↾ suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)) ∈ No )
71 2on 8098 . . . . . . . . 9 2o ∈ On
7271elexi 3463 . . . . . . . 8 2o ∈ V
7372prid2 4662 . . . . . . 7 2o ∈ {1o, 2o}
7473noextend 33287 . . . . . 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 33295 . . . . . 6 <s Or No
77 sotr2 5473 . . . . . 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 689 . . . . 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 1368 . . . 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 698 . . 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 4434 . . . . . . . 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 2848 . . . . . . 7 (∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦𝑆 = ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}))
8483dmeqd 5742 . . . . . 6 (∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 → dom 𝑆 = dom ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}))
8572dmsnop 6044 . . . . . . . 8 dom {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩} = {dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)}
8685uneq2i 4090 . . . . . . 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 5747 . . . . . . 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 6169 . . . . . . 7 suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) = (dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)})
8986, 87, 883eqtr4i 2834 . . . . . 6 dom ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}) = suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
9084, 89eqtrdi 2852 . . . . 5 (∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 → dom 𝑆 = suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦))
9190adantr 484 . . . 4 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → dom 𝑆 = suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦))
9291reseq2d 5822 . . 3 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → (𝑈 ↾ dom 𝑆) = (𝑈 ↾ suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)))
9383adantr 484 . . 3 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → 𝑆 = ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}))
9480, 92, 933brtr4d 5065 . 2 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → (𝑈 ↾ dom 𝑆) <s 𝑆)
95 simpl 486 . . 3 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → ¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
96 simpr1 1191 . . 3 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → 𝐴 No )
97 simpr2 1192 . . 3 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → 𝐴 ∈ V)
98 simpr3 1193 . . 3 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → 𝑈𝐴)
9981nosupbnd1lem6 33327 . . 3 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → (𝑈 ↾ dom 𝑆) <s 𝑆)
10095, 96, 97, 98, 99syl121anc 1372 . 2 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → (𝑈 ↾ dom 𝑆) <s 𝑆)
10194, 100pm2.61ian 811 1 ((𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴) → (𝑈 ↾ dom 𝑆) <s 𝑆)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2112  {cab 2779  wral 3109  wrex 3110  ∃!wreu 3111  ∃*wrmo 3112  Vcvv 3444  cun 3882  wss 3884  ifcif 4428  {csn 4528  cop 4534   class class class wbr 5033  cmpt 5113   Or wor 5441  dom cdm 5523  cres 5525  Rel wrel 5528  Oncon0 6163  suc csuc 6165  cio 6285  Fun wfun 6322  cfv 6328  crio 7096  1oc1o 8082  2oc2o 8083   No csur 33261   <s cslt 33262
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-ord 6166  df-on 6167  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-1o 8089  df-2o 8090  df-no 33264  df-slt 33265  df-bday 33266
This theorem is referenced by:  nosupbnd2  33330  noetalem2  33332
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