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Theorem nosupbnd1 27759

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(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2_o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))

**Assertion**
Ref	Expression
nosupbnd1	⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑈 ∈ 𝐴) → (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**
Step	Hyp	Ref	Expression
1		simpr3 1197	. . . . . 6 ⊢ ((∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑈 ∈ 𝐴)) → 𝑈 ∈ 𝐴)
2		nfv 1914	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑈 ∈ 𝐴)
3		nfcv 2905	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝐴
4		nfriota1 7395	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
5		nfcv 2905	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥 <s
6		nfcv 2905	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥𝑦
7	4, 5, 6	nfbr 5190	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦
8	7	nfn 1857	. . . . . . . . . 10 ⊢ Ⅎ𝑥 ¬ (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦
9	3, 8	nfralw 3311	. . . . . . . . 9 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 ¬ (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦
10	2, 9	nfim 1896	. . . . . . . 8 ⊢ Ⅎ𝑥((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑈 ∈ 𝐴) → ∀𝑦 ∈ 𝐴 ¬ (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦)
11		simpl 482	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑈 ∈ 𝐴)) → (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦))
12		rspe 3249	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
13	12	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑈 ∈ 𝐴)) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
14		nomaxmo 27743	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ⊆ No → ∃*𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
15	14	3ad2ant1 1134	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑈 ∈ 𝐴) → ∃*𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
16	15	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑈 ∈ 𝐴)) → ∃*𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
17		reu5 3382	. . . . . . . . . . . . 13 ⊢ (∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ↔ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ∃*𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦))
18	13, 16, 17	sylanbrc 583	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑈 ∈ 𝐴)) → ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
19		riota1 7409	. . . . . . . . . . . 12 ⊢ (∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 → ((𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ↔ (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) = 𝑥))
20	18, 19	syl 17	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑈 ∈ 𝐴)) → ((𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ↔ (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) = 𝑥))
21	11, 20	mpbid 232	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑈 ∈ 𝐴)) → (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) = 𝑥)
22		simplr 769	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑈 ∈ 𝐴)) → ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
23		nfra1 3284	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦
24		nfcv 2905	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦𝐴
25	23, 24	nfriota 7400	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦(℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
26		nfcv 2905	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦𝑥
27	25, 26	nfeq 2919	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦(℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) = 𝑥
28		breq1 5146	. . . . . . . . . . . . 13 ⊢ ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) = 𝑥 → ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦 ↔ 𝑥 <s 𝑦))
29	28	notbid 318	. . . . . . . . . . . 12 ⊢ ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) = 𝑥 → (¬ (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦 ↔ ¬ 𝑥 <s 𝑦))
30	27, 29	ralbid 3273	. . . . . . . . . . 11 ⊢ ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) = 𝑥 → (∀𝑦 ∈ 𝐴 ¬ (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦))
31	30	biimprd 248	. . . . . . . . . 10 ⊢ ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) = 𝑥 → (∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 → ∀𝑦 ∈ 𝐴 ¬ (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦))
32	21, 22, 31	sylc 65	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑈 ∈ 𝐴)) → ∀𝑦 ∈ 𝐴 ¬ (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦)
33	32	exp31 419	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 → ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑈 ∈ 𝐴) → ∀𝑦 ∈ 𝐴 ¬ (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦)))
34	10, 33	rexlimi 3259	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 → ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑈 ∈ 𝐴) → ∀𝑦 ∈ 𝐴 ¬ (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦))
35	34	imp 406	. . . . . 6 ⊢ ((∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑈 ∈ 𝐴)) → ∀𝑦 ∈ 𝐴 ¬ (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦)
36		nfcv 2905	. . . . . . . . 9 ⊢ Ⅎ𝑦 <s
37		nfcv 2905	. . . . . . . . 9 ⊢ Ⅎ𝑦𝑈
38	25, 36, 37	nfbr 5190	. . . . . . . 8 ⊢ Ⅎ𝑦(℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) <s 𝑈
39	38	nfn 1857	. . . . . . 7 ⊢ Ⅎ𝑦 ¬ (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) <s 𝑈
40		breq2 5147	. . . . . . . 8 ⊢ (𝑦 = 𝑈 → ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦 ↔ (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) <s 𝑈))
41	40	notbid 318	. . . . . . 7 ⊢ (𝑦 = 𝑈 → (¬ (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦 ↔ ¬ (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) <s 𝑈))
42	39, 41	rspc 3610	. . . . . 6 ⊢ (𝑈 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 ¬ (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) <s 𝑦 → ¬ (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) <s 𝑈))
43	1, 35, 42	sylc 65	. . . . 5 ⊢ ((∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑈 ∈ 𝐴)) → ¬ (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) <s 𝑈)
44		simpr1 1195	. . . . . . . . . 10 ⊢ ((∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑈 ∈ 𝐴)) → 𝐴 ⊆ No )
45		simpl 482	. . . . . . . . . . . 12 ⊢ ((∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑈 ∈ 𝐴)) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
46	15	adantl 481	. . . . . . . . . . . 12 ⊢ ((∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑈 ∈ 𝐴)) → ∃*𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
47	45, 46, 17	sylanbrc 583	. . . . . . . . . . 11 ⊢ ((∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑈 ∈ 𝐴)) → ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
48		riotacl 7405	. . . . . . . . . . 11 ⊢ (∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 → (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∈ 𝐴)
49	47, 48	syl 17	. . . . . . . . . 10 ⊢ ((∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑈 ∈ 𝐴)) → (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∈ 𝐴)
50	44, 49	sseldd 3984	. . . . . . . . 9 ⊢ ((∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑈 ∈ 𝐴)) → (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∈ No )
51		nofun 27694	. . . . . . . . 9 ⊢ ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∈ No → Fun (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦))
52		funrel 6583	. . . . . . . . 9 ⊢ (Fun (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) → Rel (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦))
53	50, 51, 52	3syl 18	. . . . . . . 8 ⊢ ((∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑈 ∈ 𝐴)) → Rel (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦))
54		sssucid 6464	. . . . . . . 8 ⊢ dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ⊆ suc dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
55		relssres 6040	. . . . . . . 8 ⊢ ((Rel (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∧ dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ⊆ suc dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)) → ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ↾ suc dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)) = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦))
56	53, 54, 55	sylancl 586	. . . . . . 7 ⊢ ((∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑈 ∈ 𝐴)) → ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ↾ suc dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)) = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦))
57	56	breq1d 5153	. . . . . 6 ⊢ ((∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑈 ∈ 𝐴)) → (((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ↾ suc dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)) <s (𝑈 ↾ suc dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)) ↔ (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) <s (𝑈 ↾ suc dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦))))
58	44, 1	sseldd 3984	. . . . . . 7 ⊢ ((∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑈 ∈ 𝐴)) → 𝑈 ∈ No )
59		nodmon 27695	. . . . . . . . 9 ⊢ ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∈ No → dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∈ On)
60	50, 59	syl 17	. . . . . . . 8 ⊢ ((∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑈 ∈ 𝐴)) → dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∈ On)
61		onsucb 7837	. . . . . . . 8 ⊢ (dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∈ On ↔ suc dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∈ On)
62	60, 61	sylib 218	. . . . . . 7 ⊢ ((∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑈 ∈ 𝐴)) → suc dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∈ On)
63		sltres 27707	. . . . . . 7 ⊢ (((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∈ No ∧ 𝑈 ∈ No ∧ suc dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∈ On) → (((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ↾ suc dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)) <s (𝑈 ↾ suc dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)) → (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) <s 𝑈))
64	50, 58, 62, 63	syl3anc 1373	. . . . . 6 ⊢ ((∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑈 ∈ 𝐴)) → (((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ↾ suc dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)) <s (𝑈 ↾ suc dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)) → (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) <s 𝑈))
65	57, 64	sylbird 260	. . . . 5 ⊢ ((∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑈 ∈ 𝐴)) → ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) <s (𝑈 ↾ suc dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)) → (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) <s 𝑈))
66	43, 65	mtod 198	. . . 4 ⊢ ((∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑈 ∈ 𝐴)) → ¬ (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) <s (𝑈 ↾ suc dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)))
67		noextendgt 27715	. . . . 5 ⊢ ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∈ No → (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) <s ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2_o⟩}))
68	50, 67	syl 17	. . . 4 ⊢ ((∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑈 ∈ 𝐴)) → (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) <s ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2_o⟩}))
69		noreson 27705	. . . . . 6 ⊢ ((𝑈 ∈ No ∧ suc dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∈ On) → (𝑈 ↾ suc dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)) ∈ No )
70	58, 62, 69	syl2anc 584	. . . . 5 ⊢ ((∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑈 ∈ 𝐴)) → (𝑈 ↾ suc dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)) ∈ No )
71		2on 8520	. . . . . . . . 9 ⊢ 2_o ∈ On
72	71	elexi 3503	. . . . . . . 8 ⊢ 2_o ∈ V
73	72	prid2 4763	. . . . . . 7 ⊢ 2_o ∈ {1_o, 2_o}
74	73	noextend 27711	. . . . . 6 ⊢ ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∈ No → ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2_o⟩}) ∈ No )
75	50, 74	syl 17	. . . . 5 ⊢ ((∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑈 ∈ 𝐴)) → ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2_o⟩}) ∈ No )
76		sltso 27721	. . . . . 6 ⊢ <s Or No
77		sotr2 5626	. . . . . 6 ⊢ (( <s Or No ∧ ((𝑈 ↾ suc dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)) ∈ No ∧ (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∈ No ∧ ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2_o⟩}) ∈ No )) → ((¬ (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) <s (𝑈 ↾ suc dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)) ∧ (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) <s ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2_o⟩})) → (𝑈 ↾ suc dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)) <s ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2_o⟩})))
78	76, 77	mpan 690	. . . . 5 ⊢ (((𝑈 ↾ suc dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)) ∈ No ∧ (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∈ No ∧ ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2_o⟩}) ∈ No ) → ((¬ (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) <s (𝑈 ↾ suc dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)) ∧ (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) <s ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2_o⟩})) → (𝑈 ↾ suc dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)) <s ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2_o⟩})))
79	70, 50, 75, 78	syl3anc 1373	. . . 4 ⊢ ((∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑈 ∈ 𝐴)) → ((¬ (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) <s (𝑈 ↾ suc dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)) ∧ (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) <s ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2_o⟩})) → (𝑈 ↾ suc dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)) <s ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2_o⟩})))
80	66, 68, 79	mp2and 699	. . 3 ⊢ ((∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑈 ∈ 𝐴)) → (𝑈 ↾ suc dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)) <s ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2_o⟩}))
81		nosupbnd1.1	. . . . . . . 8 ⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2_o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))
82		iftrue 4531	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 → if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2_o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) = ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2_o⟩}))
83	81, 82	eqtrid 2789	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 → 𝑆 = ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2_o⟩}))
84	83	dmeqd 5916	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 → dom 𝑆 = dom ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2_o⟩}))
85	72	dmsnop 6236	. . . . . . . 8 ⊢ dom {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2_o⟩} = {dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)}
86	85	uneq2i 4165	. . . . . . 7 ⊢ (dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ dom {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2_o⟩}) = (dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)})
87		dmun 5921	. . . . . . 7 ⊢ dom ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2_o⟩}) = (dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ dom {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2_o⟩})
88		df-suc 6390	. . . . . . 7 ⊢ suc dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) = (dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)})
89	86, 87, 88	3eqtr4i 2775	. . . . . 6 ⊢ dom ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2_o⟩}) = suc dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
90	84, 89	eqtrdi 2793	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 → dom 𝑆 = suc dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦))
91	90	adantr 480	. . . 4 ⊢ ((∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑈 ∈ 𝐴)) → dom 𝑆 = suc dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦))
92	91	reseq2d 5997	. . 3 ⊢ ((∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑈 ∈ 𝐴)) → (𝑈 ↾ dom 𝑆) = (𝑈 ↾ suc dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)))
93	83	adantr 480	. . 3 ⊢ ((∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑈 ∈ 𝐴)) → 𝑆 = ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2_o⟩}))
94	80, 92, 93	3brtr4d 5175	. 2 ⊢ ((∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑈 ∈ 𝐴)) → (𝑈 ↾ dom 𝑆) <s 𝑆)
95		simpl 482	. . 3 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑈 ∈ 𝐴)) → ¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
96		simpr1 1195	. . 3 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑈 ∈ 𝐴)) → 𝐴 ⊆ No )
97		simpr2 1196	. . 3 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑈 ∈ 𝐴)) → 𝐴 ∈ V)
98		simpr3 1197	. . 3 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑈 ∈ 𝐴)) → 𝑈 ∈ 𝐴)
99	81	nosupbnd1lem6 27758	. . 3 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ 𝑈 ∈ 𝐴) → (𝑈 ↾ dom 𝑆) <s 𝑆)
100	95, 96, 97, 98, 99	syl121anc 1377	. 2 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑈 ∈ 𝐴)) → (𝑈 ↾ dom 𝑆) <s 𝑆)
101	94, 100	pm2.61ian 812	1 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑈 ∈ 𝐴) → (𝑈 ↾ dom 𝑆) <s 𝑆)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 {cab 2714 ∀wral 3061 ∃wrex 3070 ∃!wreu 3378 ∃*wrmo 3379 Vcvv 3480 ∪ cun 3949 ⊆ wss 3951 ifcif 4525 {csn 4626 ⟨cop 4632 class class class wbr 5143 ↦ cmpt 5225 Or wor 5591 dom cdm 5685 ↾ cres 5687 Rel wrel 5690 Oncon0 6384 suc csuc 6386 ℩cio 6512 Fun wfun 6555 ‘cfv 6561 ℩crio 7387 1_oc1o 8499 2_oc2o 8500 No csur 27684 <s cslt 27685

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fo 6567 df-fv 6569 df-riota 7388 df-1o 8506 df-2o 8507 df-no 27687 df-slt 27688 df-bday 27689

This theorem is referenced by: nosupbnd2 27761 noetasuplem3 27780

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