MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nosupbnd1 Structured version   Visualization version   GIF version

Theorem nosupbnd1 27667
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 1909 . . . . . . . . 9 𝑥(𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)
3 nfcv 2899 . . . . . . . . . 10 𝑥𝐴
4 nfriota1 7389 . . . . . . . . . . . 12 𝑥(𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
5 nfcv 2899 . . . . . . . . . . . 12 𝑥 <s
6 nfcv 2899 . . . . . . . . . . . 12 𝑥𝑦
74, 5, 6nfbr 5199 . . . . . . . . . . 11 𝑥(𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦
87nfn 1852 . . . . . . . . . 10 𝑥 ¬ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦
93, 8nfralw 3306 . . . . . . . . 9 𝑥𝑦𝐴 ¬ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦
102, 9nfim 1891 . . . . . . . 8 𝑥((𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴) → ∀𝑦𝐴 ¬ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦)
11 simpl 481 . . . . . . . . . . 11 (((𝑥𝐴 ∧ ∀𝑦𝐴 ¬ 𝑥 <s 𝑦) ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → (𝑥𝐴 ∧ ∀𝑦𝐴 ¬ 𝑥 <s 𝑦))
12 rspe 3244 . . . . . . . . . . . . . 14 ((𝑥𝐴 ∧ ∀𝑦𝐴 ¬ 𝑥 <s 𝑦) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
1312adantr 479 . . . . . . . . . . . . 13 (((𝑥𝐴 ∧ ∀𝑦𝐴 ¬ 𝑥 <s 𝑦) ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
14 nomaxmo 27651 . . . . . . . . . . . . . . 15 (𝐴 No → ∃*𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
15143ad2ant1 1130 . . . . . . . . . . . . . 14 ((𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴) → ∃*𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
1615adantl 480 . . . . . . . . . . . . 13 (((𝑥𝐴 ∧ ∀𝑦𝐴 ¬ 𝑥 <s 𝑦) ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → ∃*𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
17 reu5 3376 . . . . . . . . . . . . 13 (∃!𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ↔ (∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ ∃*𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦))
1813, 16, 17sylanbrc 581 . . . . . . . . . . . 12 (((𝑥𝐴 ∧ ∀𝑦𝐴 ¬ 𝑥 <s 𝑦) ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → ∃!𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
19 riota1 7404 . . . . . . . . . . . 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 3279 . . . . . . . . . . . . . 14 𝑦𝑦𝐴 ¬ 𝑥 <s 𝑦
24 nfcv 2899 . . . . . . . . . . . . . 14 𝑦𝐴
2523, 24nfriota 7395 . . . . . . . . . . . . 13 𝑦(𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
26 nfcv 2899 . . . . . . . . . . . . 13 𝑦𝑥
2725, 26nfeq 2913 . . . . . . . . . . . 12 𝑦(𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) = 𝑥
28 breq1 5155 . . . . . . . . . . . . 13 ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) = 𝑥 → ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦𝑥 <s 𝑦))
2928notbid 317 . . . . . . . . . . . 12 ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) = 𝑥 → (¬ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦 ↔ ¬ 𝑥 <s 𝑦))
3027, 29ralbid 3268 . . . . . . . . . . 11 ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) = 𝑥 → (∀𝑦𝐴 ¬ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦 ↔ ∀𝑦𝐴 ¬ 𝑥 <s 𝑦))
3130biimprd 247 . . . . . . . . . 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 418 . . . . . . . 8 (𝑥𝐴 → (∀𝑦𝐴 ¬ 𝑥 <s 𝑦 → ((𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴) → ∀𝑦𝐴 ¬ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦)))
3410, 33rexlimi 3254 . . . . . . 7 (∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 → ((𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴) → ∀𝑦𝐴 ¬ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦))
3534imp 405 . . . . . 6 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → ∀𝑦𝐴 ¬ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦)
36 nfcv 2899 . . . . . . . . 9 𝑦 <s
37 nfcv 2899 . . . . . . . . 9 𝑦𝑈
3825, 36, 37nfbr 5199 . . . . . . . 8 𝑦(𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑈
3938nfn 1852 . . . . . . 7 𝑦 ¬ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑈
40 breq2 5156 . . . . . . . 8 (𝑦 = 𝑈 → ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦 ↔ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑈))
4140notbid 317 . . . . . . 7 (𝑦 = 𝑈 → (¬ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦 ↔ ¬ (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) <s 𝑈))
4239, 41rspc 3599 . . . . . 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 481 . . . . . . . . . . . 12 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
4615adantl 480 . . . . . . . . . . . 12 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → ∃*𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
4745, 46, 17sylanbrc 581 . . . . . . . . . . 11 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → ∃!𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
48 riotacl 7400 . . . . . . . . . . 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 3983 . . . . . . . . 9 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∈ No )
51 nofun 27602 . . . . . . . . 9 ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∈ No → Fun (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦))
52 funrel 6575 . . . . . . . . 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 6454 . . . . . . . 8 dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ⊆ suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
55 relssres 6031 . . . . . . . 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 584 . . . . . . 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 5162 . . . . . 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 3983 . . . . . . 7 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → 𝑈 No )
59 nodmon 27603 . . . . . . . . 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 onsucb 7826 . . . . . . . 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 27615 . . . . . . 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 259 . . . . 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 27623 . . . . 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 27613 . . . . . 6 ((𝑈 No ∧ suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∈ On) → (𝑈 ↾ suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)) ∈ No )
7058, 62, 69syl2anc 582 . . . . 5 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → (𝑈 ↾ suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)) ∈ No )
71 2on 8507 . . . . . . . . 9 2o ∈ On
7271elexi 3493 . . . . . . . 8 2o ∈ V
7372prid2 4772 . . . . . . 7 2o ∈ {1o, 2o}
7473noextend 27619 . . . . . 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 27629 . . . . . 6 <s Or No
77 sotr2 5626 . . . . . 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 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 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 4538 . . . . . . . 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 2780 . . . . . . 7 (∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦𝑆 = ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}))
8483dmeqd 5912 . . . . . 6 (∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 → dom 𝑆 = dom ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}))
8572dmsnop 6225 . . . . . . . 8 dom {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩} = {dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)}
8685uneq2i 4161 . . . . . . 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 5917 . . . . . . 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 6380 . . . . . . 7 suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) = (dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)})
8986, 87, 883eqtr4i 2766 . . . . . 6 dom ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}) = suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
9084, 89eqtrdi 2784 . . . . 5 (∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 → dom 𝑆 = suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦))
9190adantr 479 . . . 4 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → dom 𝑆 = suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦))
9291reseq2d 5989 . . 3 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → (𝑈 ↾ dom 𝑆) = (𝑈 ↾ suc dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)))
9383adantr 479 . . 3 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → 𝑆 = ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}))
9480, 92, 933brtr4d 5184 . 2 ((∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V ∧ 𝑈𝐴)) → (𝑈 ↾ dom 𝑆) <s 𝑆)
95 simpl 481 . . 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 27666 . . 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 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 394  w3a 1084   = wceq 1533  wcel 2098  {cab 2705  wral 3058  wrex 3067  ∃!wreu 3372  ∃*wrmo 3373  Vcvv 3473  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 8486  2oc2o 8487   No csur 27593   <s cslt 27594
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 7746
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 8493  df-2o 8494  df-no 27596  df-slt 27597  df-bday 27598
This theorem is referenced by:  nosupbnd2  27669  noetasuplem3  27688
  Copyright terms: Public domain W3C validator