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Theorem nosupbnd1 26967
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 1196 . . . . . 6 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → 𝑈𝐴)
2 nfv 1917 . . . . . . . . 9 𝑥(𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)
3 nfcv 2905 . . . . . . . . . 10 𝑥𝐴
4 nfriota1 7304 . . . . . . . . . . . 12 𝑥(𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
5 nfcv 2905 . . . . . . . . . . . 12 𝑥 <s
6 nfcv 2905 . . . . . . . . . . . 12 𝑥𝑦
74, 5, 6nfbr 5143 . . . . . . . . . . 11 𝑥(𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦
87nfn 1860 . . . . . . . . . 10 𝑥 ¬ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦
93, 8nfralw 3291 . . . . . . . . 9 𝑥𝑦𝐴 ¬ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦
102, 9nfim 1899 . . . . . . . 8 𝑥((𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴) → ∀𝑦𝐴 ¬ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦)
11 simpl 484 . . . . . . . . . . 11 (((𝑥𝐴 ∧ ∀𝑦𝐴 ¬ 𝑥 <s 𝑦) ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → (𝑥𝐴 ∧ ∀𝑦𝐴 ¬ 𝑥 <s 𝑦))
12 rspe 3229 . . . . . . . . . . . . . 14 ((𝑥𝐴 ∧ ∀𝑦𝐴 ¬ 𝑥 <s 𝑦) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
1312adantr 482 . . . . . . . . . . . . 13 (((𝑥𝐴 ∧ ∀𝑦𝐴 ¬ 𝑥 <s 𝑦) ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
14 nomaxmo 26951 . . . . . . . . . . . . . . 15 (𝐴 No → ∃*𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
15143ad2ant1 1133 . . . . . . . . . . . . . 14 ((𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴) → ∃*𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
1615adantl 483 . . . . . . . . . . . . 13 (((𝑥𝐴 ∧ ∀𝑦𝐴 ¬ 𝑥 <s 𝑦) ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → ∃*𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
17 reu5 3352 . . . . . . . . . . . . 13 (∃!𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ↔ (∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ∃*𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦))
1813, 16, 17sylanbrc 584 . . . . . . . . . . . 12 (((𝑥𝐴 ∧ ∀𝑦𝐴 ¬ 𝑥 <s 𝑦) ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → ∃!𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
19 riota1 7319 . . . . . . . . . . . 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 231 . . . . . . . . . 10 (((𝑥𝐴 ∧ ∀𝑦𝐴 ¬ 𝑥 <s 𝑦) ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) = 𝑥)
22 simplr 767 . . . . . . . . . 10 (((𝑥𝐴 ∧ ∀𝑦𝐴 ¬ 𝑥 <s 𝑦) ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → ∀𝑦𝐴 ¬ 𝑥 <s 𝑦)
23 nfra1 3264 . . . . . . . . . . . . . 14 𝑦𝑦𝐴 ¬ 𝑥 <s 𝑦
24 nfcv 2905 . . . . . . . . . . . . . 14 𝑦𝐴
2523, 24nfriota 7310 . . . . . . . . . . . . 13 𝑦(𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
26 nfcv 2905 . . . . . . . . . . . . 13 𝑦𝑥
2725, 26nfeq 2918 . . . . . . . . . . . 12 𝑦(𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) = 𝑥
28 breq1 5099 . . . . . . . . . . . . 13 ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) = 𝑥 → ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦𝑥 <s 𝑦))
2928notbid 318 . . . . . . . . . . . 12 ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) = 𝑥 → (¬ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦 ↔ ¬ 𝑥 <s 𝑦))
3027, 29ralbid 3253 . . . . . . . . . . 11 ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) = 𝑥 → (∀𝑦𝐴 ¬ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦 ↔ ∀𝑦𝐴 ¬ 𝑥 <s 𝑦))
3130biimprd 248 . . . . . . . . . 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 421 . . . . . . . 8 (𝑥𝐴 → (∀𝑦𝐴 ¬ 𝑥 <s 𝑦 → ((𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴) → ∀𝑦𝐴 ¬ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦)))
3410, 33rexlimi 3239 . . . . . . 7 (∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 → ((𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴) → ∀𝑦𝐴 ¬ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦))
3534imp 408 . . . . . 6 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → ∀𝑦𝐴 ¬ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦)
36 nfcv 2905 . . . . . . . . 9 𝑦 <s
37 nfcv 2905 . . . . . . . . 9 𝑦𝑈
3825, 36, 37nfbr 5143 . . . . . . . 8 𝑦(𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑈
3938nfn 1860 . . . . . . 7 𝑦 ¬ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑈
40 breq2 5100 . . . . . . . 8 (𝑦 = 𝑈 → ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦 ↔ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑈))
4140notbid 318 . . . . . . 7 (𝑦 = 𝑈 → (¬ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦 ↔ ¬ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑈))
4239, 41rspc 3561 . . . . . 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 1194 . . . . . . . . . 10 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → 𝐴 No )
45 simpl 484 . . . . . . . . . . . 12 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
4615adantl 483 . . . . . . . . . . . 12 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → ∃*𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
4745, 46, 17sylanbrc 584 . . . . . . . . . . 11 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → ∃!𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
48 riotacl 7315 . . . . . . . . . . 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 3936 . . . . . . . . 9 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∈ No )
51 nofun 26902 . . . . . . . . 9 ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∈ No → Fun (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦))
52 funrel 6505 . . . . . . . . 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 6385 . . . . . . . 8 dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ⊆ suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
55 relssres 5968 . . . . . . . 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 587 . . . . . . 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 5106 . . . . . 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 3936 . . . . . . 7 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → 𝑈 No )
59 nodmon 26903 . . . . . . . . 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 7734 . . . . . . . 8 (dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∈ On ↔ suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∈ On)
6260, 61sylib 217 . . . . . . 7 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∈ On)
63 sltres 26915 . . . . . . 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 1371 . . . . . 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 260 . . . . 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 197 . . . 4 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → ¬ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s (𝑈 ↾ suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)))
67 noextendgt 26923 . . . . 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 26913 . . . . . 6 ((𝑈 No ∧ suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∈ On) → (𝑈 ↾ suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)) ∈ No )
7058, 62, 69syl2anc 585 . . . . 5 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → (𝑈 ↾ suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)) ∈ No )
71 2on 8385 . . . . . . . . 9 2o ∈ On
7271elexi 3461 . . . . . . . 8 2o ∈ V
7372prid2 4715 . . . . . . 7 2o ∈ {1o, 2o}
7473noextend 26919 . . . . . 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 26929 . . . . . 6 <s Or No
77 sotr2 5568 . . . . . 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 688 . . . . 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 1371 . . . 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 697 . . 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 4483 . . . . . . . 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, 82eqtrid 2789 . . . . . . 7 (∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦𝑆 = ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}))
8483dmeqd 5851 . . . . . 6 (∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 → dom 𝑆 = dom ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}))
8572dmsnop 6158 . . . . . . . 8 dom {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩} = {dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)}
8685uneq2i 4111 . . . . . . 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 5856 . . . . . . 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 6312 . . . . . . 7 suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) = (dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)})
8986, 87, 883eqtr4i 2775 . . . . . 6 dom ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}) = suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
9084, 89eqtrdi 2793 . . . . 5 (∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 → dom 𝑆 = suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦))
9190adantr 482 . . . 4 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → dom 𝑆 = suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦))
9291reseq2d 5927 . . 3 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → (𝑈 ↾ dom 𝑆) = (𝑈 ↾ suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)))
9383adantr 482 . . 3 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → 𝑆 = ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}))
9480, 92, 933brtr4d 5128 . 2 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → (𝑈 ↾ dom 𝑆) <s 𝑆)
95 simpl 484 . . 3 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → ¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
96 simpr1 1194 . . 3 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → 𝐴 No )
97 simpr2 1195 . . 3 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → 𝐴 ∈ V)
98 simpr3 1196 . . 3 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → 𝑈𝐴)
9981nosupbnd1lem6 26966 . . 3 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → (𝑈 ↾ dom 𝑆) <s 𝑆)
10095, 96, 97, 98, 99syl121anc 1375 . 2 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → (𝑈 ↾ dom 𝑆) <s 𝑆)
10194, 100pm2.61ian 810 1 ((𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴) → (𝑈 ↾ dom 𝑆) <s 𝑆)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  w3a 1087   = wceq 1541  wcel 2106  {cab 2714  wral 3062  wrex 3071  ∃!wreu 3348  ∃*wrmo 3349  Vcvv 3442  cun 3899  wss 3901  ifcif 4477  {csn 4577  cop 4583   class class class wbr 5096  cmpt 5179   Or wor 5535  dom cdm 5624  cres 5626  Rel wrel 5629  Oncon0 6306  suc csuc 6308  cio 6433  Fun wfun 6477  cfv 6483  crio 7296  1oc1o 8364  2oc2o 8365   No csur 26893   <s cslt 26894
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5233  ax-sep 5247  ax-nul 5254  ax-pr 5376  ax-un 7654
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3350  df-reu 3351  df-rab 3405  df-v 3444  df-sbc 3731  df-csb 3847  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3920  df-nul 4274  df-if 4478  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-uni 4857  df-int 4899  df-iun 4947  df-br 5097  df-opab 5159  df-mpt 5180  df-tr 5214  df-id 5522  df-eprel 5528  df-po 5536  df-so 5537  df-fr 5579  df-we 5581  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ord 6309  df-on 6310  df-suc 6312  df-iota 6435  df-fun 6485  df-fn 6486  df-f 6487  df-f1 6488  df-fo 6489  df-f1o 6490  df-fv 6491  df-riota 7297  df-1o 8371  df-2o 8372  df-no 26896  df-slt 26897  df-bday 26898
This theorem is referenced by:  nosupbnd2  26969  noetasuplem3  26988
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