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Theorem nosupbnd2 27880

Description: Bounding law from above for the surreal supremum. Proposition 4.3 of [Lipparini] p. 6. (Contributed by Scott Fenton, 6-Dec-2021.)

**Hypothesis**
Ref	Expression
nosupbnd2.1	⊢ 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
nosupbnd2	⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) → (∀𝑎 ∈ 𝐴 𝑎 <s 𝑍 ↔ ¬ (𝑍 ↾ dom 𝑆) <s 𝑆))

Distinct variable groups: 𝐴,𝑎,𝑔,𝑢,𝑣,𝑥,𝑦 𝑍,𝑎,𝑔,𝑥 𝑆,𝑎,𝑔 Allowed substitution hints: 𝑆(𝑥,𝑦,𝑣,𝑢) 𝑍(𝑦,𝑣,𝑢)

Proof of Theorem nosupbnd2 Dummy variables 𝑝 𝑞 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1944	. . . . . 6 ⊢ Ⅎ𝑥((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)
2		nfcv 2925	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑍
3		nosupbnd2.1	. . . . . . . . . . 11 ⊢ 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))))
4		nfre1 3290	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦
5		nfriota1 7374	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
6	5	nfdm 5941	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
7		nfcv 2925	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥2_o
8	6, 7	nfop 4854	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2_o⟩
9	8	nfsn 4673	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥{⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2_o⟩}
10	5, 9	nfun 4124	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2_o⟩})
11		nfcv 2925	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥{𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}
12		nfiota1 6494	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))
13	11, 12	nfmpt 5209	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(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)))
14	4, 10, 13	nfif 4518	. . . . . . . . . . 11 ⊢ Ⅎ𝑥if(∃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))))
15	3, 14	nfcxfr 2923	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑆
16	15	nfdm 5941	. . . . . . . . 9 ⊢ Ⅎ𝑥dom 𝑆
17	2, 16	nfres 5980	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑍 ↾ dom 𝑆)
18		nfcv 2925	. . . . . . . 8 ⊢ Ⅎ𝑥 <s
19	17, 18, 15	nfbr 5158	. . . . . . 7 ⊢ Ⅎ𝑥(𝑍 ↾ dom 𝑆) <s 𝑆
20	19	nfn 1887	. . . . . 6 ⊢ Ⅎ𝑥 ¬ (𝑍 ↾ dom 𝑆) <s 𝑆
21	1, 20	nfim 1926	. . . . 5 ⊢ Ⅎ𝑥(((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) → ¬ (𝑍 ↾ dom 𝑆) <s 𝑆)
22		simpl 487	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) → (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦))
23		rspe 3255	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
24	23	adantr 485	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
25		nomaxmo 27862	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ No → ∃*𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
26	25	3ad2ant1 1151	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) → ∃*𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
27	26	ad2antrl 740	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) → ∃*𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
28		reu5 3371	. . . . . . . . . . 11 ⊢ (∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ↔ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ∃*𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦))
29	24, 27, 28	sylanbrc 594	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) → ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
30		riota1 7388	. . . . . . . . . 10 ⊢ (∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 → ((𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ↔ (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) = 𝑥))
31	29, 30	syl 18	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) → ((𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ↔ (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) = 𝑥))
32	22, 31	mpbid 235	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) → (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) = 𝑥)
33		nosupbnd2lem1 27879	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) → ¬ (𝑍 ↾ suc dom 𝑥) <s (𝑥 ∪ {⟨dom 𝑥, 2_o⟩}))
34	33	3expb 1138	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) → ¬ (𝑍 ↾ suc dom 𝑥) <s (𝑥 ∪ {⟨dom 𝑥, 2_o⟩}))
35		dmeq 5893	. . . . . . . . . . . . 13 ⊢ ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) = 𝑥 → dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) = dom 𝑥)
36	35	suceqd 6428	. . . . . . . . . . . 12 ⊢ ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) = 𝑥 → suc dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) = suc dom 𝑥)
37	36	reseq2d 5978	. . . . . . . . . . 11 ⊢ ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) = 𝑥 → (𝑍 ↾ suc dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)) = (𝑍 ↾ suc dom 𝑥))
38		id 23	. . . . . . . . . . . 12 ⊢ ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) = 𝑥 → (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) = 𝑥)
39	35	opeq1d 4844	. . . . . . . . . . . . 13 ⊢ ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) = 𝑥 → ⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2_o⟩ = ⟨dom 𝑥, 2_o⟩)
40	39	sneqd 4601	. . . . . . . . . . . 12 ⊢ ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) = 𝑥 → {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2_o⟩} = {⟨dom 𝑥, 2_o⟩})
41	38, 40	uneq12d 4123	. . . . . . . . . . 11 ⊢ ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) = 𝑥 → ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2_o⟩}) = (𝑥 ∪ {⟨dom 𝑥, 2_o⟩}))
42	37, 41	breq12d 5122	. . . . . . . . . 10 ⊢ ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) = 𝑥 → ((𝑍 ↾ 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⟩}) ↔ (𝑍 ↾ suc dom 𝑥) <s (𝑥 ∪ {⟨dom 𝑥, 2_o⟩})))
43	42	notbid 321	. . . . . . . . 9 ⊢ ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) = 𝑥 → (¬ (𝑍 ↾ 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⟩}) ↔ ¬ (𝑍 ↾ suc dom 𝑥) <s (𝑥 ∪ {⟨dom 𝑥, 2_o⟩})))
44	34, 43	syl5ibrcom 250	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) → ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) = 𝑥 → ¬ (𝑍 ↾ 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⟩})))
45	32, 44	mpd 16	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) → ¬ (𝑍 ↾ 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⟩}))
46		iftrue 4493	. . . . . . . . . . . . . 14 ⊢ (∃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⟩}))
47	3, 46	eqtrid 2810	. . . . . . . . . . . . 13 ⊢ (∃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⟩}))
48	23, 47	syl 18	. . . . . . . . . . . 12 ⊢ ((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⟩}))
49	48	dmeqd 5895	. . . . . . . . . . 11 ⊢ ((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⟩}))
50		2on 8463	. . . . . . . . . . . . . . 15 ⊢ 2_o ∈ On
51	50	elexi 3477	. . . . . . . . . . . . . 14 ⊢ 2_o ∈ V
52	51	dmsnop 6217	. . . . . . . . . . . . 13 ⊢ dom {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2_o⟩} = {dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)}
53	52	uneq2i 4119	. . . . . . . . . . . 12 ⊢ (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)})
54		dmun 5900	. . . . . . . . . . . 12 ⊢ 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⟩})
55		df-suc 6366	. . . . . . . . . . . 12 ⊢ 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)})
56	53, 54, 55	3eqtr4i 2796	. . . . . . . . . . 11 ⊢ 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)
57	49, 56	eqtrdi 2814	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) → dom 𝑆 = suc dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦))
58	57	reseq2d 5978	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) → (𝑍 ↾ dom 𝑆) = (𝑍 ↾ suc dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)))
59	58	adantr 485	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) → (𝑍 ↾ dom 𝑆) = (𝑍 ↾ suc dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)))
60	48	adantr 485	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) → 𝑆 = ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2_o⟩}))
61	59, 60	breq12d 5122	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) → ((𝑍 ↾ dom 𝑆) <s 𝑆 ↔ (𝑍 ↾ 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⟩})))
62	45, 61	mtbird 328	. . . . . 6 ⊢ (((𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) → ¬ (𝑍 ↾ dom 𝑆) <s 𝑆)
63	62	exp31 424	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 → (((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) → ¬ (𝑍 ↾ dom 𝑆) <s 𝑆)))
64	21, 63	rexlimi 3265	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 → (((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) → ¬ (𝑍 ↾ dom 𝑆) <s 𝑆))
65	64	imp 411	. . 3 ⊢ ((∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) → ¬ (𝑍 ↾ dom 𝑆) <s 𝑆)
66	3	nosupno 27867	. . . . . . . 8 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V) → 𝑆 ∈ No )
67	66	3adant3 1150	. . . . . . 7 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) → 𝑆 ∈ No )
68	67	ad2antrl 740	. . . . . 6 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) → 𝑆 ∈ No )
69		nodmon 27814	. . . . . . 7 ⊢ (𝑆 ∈ No → dom 𝑆 ∈ On)
70	68, 69	syl 18	. . . . . 6 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) → dom 𝑆 ∈ On)
71		noreson 27824	. . . . . 6 ⊢ ((𝑆 ∈ No ∧ dom 𝑆 ∈ On) → (𝑆 ↾ dom 𝑆) ∈ No )
72	68, 70, 71	syl2anc 595	. . . . 5 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) → (𝑆 ↾ dom 𝑆) ∈ No )
73		simprl3 1239	. . . . . 6 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) → 𝑍 ∈ No )
74		noreson 27824	. . . . . 6 ⊢ ((𝑍 ∈ No ∧ dom 𝑆 ∈ On) → (𝑍 ↾ dom 𝑆) ∈ No )
75	73, 70, 74	syl2anc 595	. . . . 5 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) → (𝑍 ↾ dom 𝑆) ∈ No )
76		dmres 6011	. . . . . . 7 ⊢ dom (𝑆 ↾ dom 𝑆) = (dom 𝑆 ∩ dom 𝑆)
77		inss2 4190	. . . . . . 7 ⊢ (dom 𝑆 ∩ dom 𝑆) ⊆ dom 𝑆
78	76, 77	eqsstri 3983	. . . . . 6 ⊢ dom (𝑆 ↾ dom 𝑆) ⊆ dom 𝑆
79	78	a1i 11	. . . . 5 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) → dom (𝑆 ↾ dom 𝑆) ⊆ dom 𝑆)
80		dmres 6011	. . . . . . 7 ⊢ dom (𝑍 ↾ dom 𝑆) = (dom 𝑆 ∩ dom 𝑍)
81		inss1 4189	. . . . . . 7 ⊢ (dom 𝑆 ∩ dom 𝑍) ⊆ dom 𝑆
82	80, 81	eqsstri 3983	. . . . . 6 ⊢ dom (𝑍 ↾ dom 𝑆) ⊆ dom 𝑆
83	82	a1i 11	. . . . 5 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) → dom (𝑍 ↾ dom 𝑆) ⊆ dom 𝑆)
84	3	nosupdm 27868	. . . . . . . . . . 11 ⊢ (¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 → dom 𝑆 = {𝑔 ∣ ∃𝑝 ∈ 𝐴 (𝑔 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐴 (¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑔) = (𝑞 ↾ suc 𝑔)))})
85	84	eqabrd 2904	. . . . . . . . . 10 ⊢ (¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 → (𝑔 ∈ dom 𝑆 ↔ ∃𝑝 ∈ 𝐴 (𝑔 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐴 (¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑔) = (𝑞 ↾ suc 𝑔)))))
86	85	adantr 485	. . . . . . . . 9 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) → (𝑔 ∈ dom 𝑆 ↔ ∃𝑝 ∈ 𝐴 (𝑔 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐴 (¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑔) = (𝑞 ↾ suc 𝑔)))))
87		simprl 782	. . . . . . . . . . . . . 14 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) ∧ (𝑝 ∈ 𝐴 ∧ (𝑔 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐴 (¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑔) = (𝑞 ↾ suc 𝑔))))) → 𝑝 ∈ 𝐴)
88		simplrr 789	. . . . . . . . . . . . . 14 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) ∧ (𝑝 ∈ 𝐴 ∧ (𝑔 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐴 (¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑔) = (𝑞 ↾ suc 𝑔))))) → ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)
89		breq1 5112	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑝 → (𝑎 <s 𝑍 ↔ 𝑝 <s 𝑍))
90	89	rspcv 3577	. . . . . . . . . . . . . 14 ⊢ (𝑝 ∈ 𝐴 → (∀𝑎 ∈ 𝐴 𝑎 <s 𝑍 → 𝑝 <s 𝑍))
91	87, 88, 90	sylc 66	. . . . . . . . . . . . 13 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) ∧ (𝑝 ∈ 𝐴 ∧ (𝑔 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐴 (¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑔) = (𝑞 ↾ suc 𝑔))))) → 𝑝 <s 𝑍)
92		simprl1 1237	. . . . . . . . . . . . . . . 16 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) → 𝐴 ⊆ No )
93	92	adantr 485	. . . . . . . . . . . . . . 15 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) ∧ (𝑝 ∈ 𝐴 ∧ (𝑔 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐴 (¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑔) = (𝑞 ↾ suc 𝑔))))) → 𝐴 ⊆ No )
94	93, 87	sseldd 3938	. . . . . . . . . . . . . 14 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) ∧ (𝑝 ∈ 𝐴 ∧ (𝑔 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐴 (¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑔) = (𝑞 ↾ suc 𝑔))))) → 𝑝 ∈ No )
95	73	adantr 485	. . . . . . . . . . . . . 14 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) ∧ (𝑝 ∈ 𝐴 ∧ (𝑔 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐴 (¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑔) = (𝑞 ↾ suc 𝑔))))) → 𝑍 ∈ No )
96		ltsso 27840	. . . . . . . . . . . . . . 15 ⊢ <s Or No
97		soasym 5602	. . . . . . . . . . . . . . 15 ⊢ (( <s Or No ∧ (𝑝 ∈ No ∧ 𝑍 ∈ No )) → (𝑝 <s 𝑍 → ¬ 𝑍 <s 𝑝))
98	96, 97	mpan 702	. . . . . . . . . . . . . 14 ⊢ ((𝑝 ∈ No ∧ 𝑍 ∈ No ) → (𝑝 <s 𝑍 → ¬ 𝑍 <s 𝑝))
99	94, 95, 98	syl2anc 595	. . . . . . . . . . . . 13 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) ∧ (𝑝 ∈ 𝐴 ∧ (𝑔 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐴 (¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑔) = (𝑞 ↾ suc 𝑔))))) → (𝑝 <s 𝑍 → ¬ 𝑍 <s 𝑝))
100	91, 99	mpd 16	. . . . . . . . . . . 12 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) ∧ (𝑝 ∈ 𝐴 ∧ (𝑔 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐴 (¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑔) = (𝑞 ↾ suc 𝑔))))) → ¬ 𝑍 <s 𝑝)
101		nodmon 27814	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 ∈ No → dom 𝑝 ∈ On)
102	94, 101	syl 18	. . . . . . . . . . . . . . 15 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) ∧ (𝑝 ∈ 𝐴 ∧ (𝑔 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐴 (¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑔) = (𝑞 ↾ suc 𝑔))))) → dom 𝑝 ∈ On)
103		simprrl 792	. . . . . . . . . . . . . . 15 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) ∧ (𝑝 ∈ 𝐴 ∧ (𝑔 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐴 (¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑔) = (𝑞 ↾ suc 𝑔))))) → 𝑔 ∈ dom 𝑝)
104		onelon 6385	. . . . . . . . . . . . . . 15 ⊢ ((dom 𝑝 ∈ On ∧ 𝑔 ∈ dom 𝑝) → 𝑔 ∈ On)
105	102, 103, 104	syl2anc 595	. . . . . . . . . . . . . 14 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) ∧ (𝑝 ∈ 𝐴 ∧ (𝑔 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐴 (¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑔) = (𝑞 ↾ suc 𝑔))))) → 𝑔 ∈ On)
106		onsucb 7809	. . . . . . . . . . . . . 14 ⊢ (𝑔 ∈ On ↔ suc 𝑔 ∈ On)
107	105, 106	sylib 221	. . . . . . . . . . . . 13 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) ∧ (𝑝 ∈ 𝐴 ∧ (𝑔 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐴 (¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑔) = (𝑞 ↾ suc 𝑔))))) → suc 𝑔 ∈ On)
108		ltsres 27826	. . . . . . . . . . . . 13 ⊢ ((𝑍 ∈ No ∧ 𝑝 ∈ No ∧ suc 𝑔 ∈ On) → ((𝑍 ↾ suc 𝑔) <s (𝑝 ↾ suc 𝑔) → 𝑍 <s 𝑝))
109	95, 94, 107, 108	syl3anc 1398	. . . . . . . . . . . 12 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) ∧ (𝑝 ∈ 𝐴 ∧ (𝑔 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐴 (¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑔) = (𝑞 ↾ suc 𝑔))))) → ((𝑍 ↾ suc 𝑔) <s (𝑝 ↾ suc 𝑔) → 𝑍 <s 𝑝))
110	100, 109	mtod 201	. . . . . . . . . . 11 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) ∧ (𝑝 ∈ 𝐴 ∧ (𝑔 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐴 (¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑔) = (𝑞 ↾ suc 𝑔))))) → ¬ (𝑍 ↾ suc 𝑔) <s (𝑝 ↾ suc 𝑔))
111		simpll 778	. . . . . . . . . . . . 13 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) ∧ (𝑝 ∈ 𝐴 ∧ (𝑔 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐴 (¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑔) = (𝑞 ↾ suc 𝑔))))) → ¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
112		simprl2 1238	. . . . . . . . . . . . . . 15 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) → 𝐴 ∈ V)
113	92, 112	jca 520	. . . . . . . . . . . . . 14 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) → (𝐴 ⊆ No ∧ 𝐴 ∈ V))
114	113	adantr 485	. . . . . . . . . . . . 13 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) ∧ (𝑝 ∈ 𝐴 ∧ (𝑔 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐴 (¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑔) = (𝑞 ↾ suc 𝑔))))) → (𝐴 ⊆ No ∧ 𝐴 ∈ V))
115		simprrr 793	. . . . . . . . . . . . . 14 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) ∧ (𝑝 ∈ 𝐴 ∧ (𝑔 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐴 (¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑔) = (𝑞 ↾ suc 𝑔))))) → ∀𝑞 ∈ 𝐴 (¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑔) = (𝑞 ↾ suc 𝑔)))
116		breq1 5112	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = 𝑞 → (𝑣 <s 𝑝 ↔ 𝑞 <s 𝑝))
117	116	notbid 321	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 𝑞 → (¬ 𝑣 <s 𝑝 ↔ ¬ 𝑞 <s 𝑝))
118		reseq1 5972	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = 𝑞 → (𝑣 ↾ suc 𝑔) = (𝑞 ↾ suc 𝑔))
119	118	eqeq2d 2774	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 𝑞 → ((𝑝 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔) ↔ (𝑝 ↾ suc 𝑔) = (𝑞 ↾ suc 𝑔)))
120	117, 119	imbi12d 347	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = 𝑞 → ((¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ↔ (¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑔) = (𝑞 ↾ suc 𝑔))))
121	120	cbvralvw 3243	. . . . . . . . . . . . . 14 ⊢ (∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ↔ ∀𝑞 ∈ 𝐴 (¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑔) = (𝑞 ↾ suc 𝑔)))
122	115, 121	sylibr 237	. . . . . . . . . . . . 13 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) ∧ (𝑝 ∈ 𝐴 ∧ (𝑔 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐴 (¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑔) = (𝑞 ↾ suc 𝑔))))) → ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)))
123	3	nosupres 27871	. . . . . . . . . . . . 13 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑝 ∈ 𝐴 ∧ 𝑔 ∈ dom 𝑝 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)))) → (𝑆 ↾ suc 𝑔) = (𝑝 ↾ suc 𝑔))
124	111, 114, 87, 103, 122, 123	syl113anc 1409	. . . . . . . . . . . 12 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) ∧ (𝑝 ∈ 𝐴 ∧ (𝑔 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐴 (¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑔) = (𝑞 ↾ suc 𝑔))))) → (𝑆 ↾ suc 𝑔) = (𝑝 ↾ suc 𝑔))
125	124	breq2d 5121	. . . . . . . . . . 11 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) ∧ (𝑝 ∈ 𝐴 ∧ (𝑔 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐴 (¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑔) = (𝑞 ↾ suc 𝑔))))) → ((𝑍 ↾ suc 𝑔) <s (𝑆 ↾ suc 𝑔) ↔ (𝑍 ↾ suc 𝑔) <s (𝑝 ↾ suc 𝑔)))
126	110, 125	mtbird 328	. . . . . . . . . 10 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) ∧ (𝑝 ∈ 𝐴 ∧ (𝑔 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐴 (¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑔) = (𝑞 ↾ suc 𝑔))))) → ¬ (𝑍 ↾ suc 𝑔) <s (𝑆 ↾ suc 𝑔))
127	126	rexlimdvaa 3167	. . . . . . . . 9 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) → (∃𝑝 ∈ 𝐴 (𝑔 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐴 (¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑔) = (𝑞 ↾ suc 𝑔))) → ¬ (𝑍 ↾ suc 𝑔) <s (𝑆 ↾ suc 𝑔)))
128	86, 127	sylbid 243	. . . . . . . 8 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) → (𝑔 ∈ dom 𝑆 → ¬ (𝑍 ↾ suc 𝑔) <s (𝑆 ↾ suc 𝑔)))
129	128	imp 411	. . . . . . 7 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) ∧ 𝑔 ∈ dom 𝑆) → ¬ (𝑍 ↾ suc 𝑔) <s (𝑆 ↾ suc 𝑔))
130	68	adantr 485	. . . . . . . . . . 11 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) ∧ 𝑔 ∈ dom 𝑆) → 𝑆 ∈ No )
131		nodmord 27817	. . . . . . . . . . 11 ⊢ (𝑆 ∈ No → Ord dom 𝑆)
132	130, 131	syl 18	. . . . . . . . . 10 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) ∧ 𝑔 ∈ dom 𝑆) → Ord dom 𝑆)
133		simpr 489	. . . . . . . . . 10 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) ∧ 𝑔 ∈ dom 𝑆) → 𝑔 ∈ dom 𝑆)
134		ordsucss 7810	. . . . . . . . . 10 ⊢ (Ord dom 𝑆 → (𝑔 ∈ dom 𝑆 → suc 𝑔 ⊆ dom 𝑆))
135	132, 133, 134	sylc 66	. . . . . . . . 9 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) ∧ 𝑔 ∈ dom 𝑆) → suc 𝑔 ⊆ dom 𝑆)
136	135	resabs1d 6007	. . . . . . . 8 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) ∧ 𝑔 ∈ dom 𝑆) → ((𝑍 ↾ dom 𝑆) ↾ suc 𝑔) = (𝑍 ↾ suc 𝑔))
137	135	resabs1d 6007	. . . . . . . 8 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) ∧ 𝑔 ∈ dom 𝑆) → ((𝑆 ↾ dom 𝑆) ↾ suc 𝑔) = (𝑆 ↾ suc 𝑔))
138	136, 137	breq12d 5122	. . . . . . 7 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) ∧ 𝑔 ∈ dom 𝑆) → (((𝑍 ↾ dom 𝑆) ↾ suc 𝑔) <s ((𝑆 ↾ dom 𝑆) ↾ suc 𝑔) ↔ (𝑍 ↾ suc 𝑔) <s (𝑆 ↾ suc 𝑔)))
139	129, 138	mtbird 328	. . . . . 6 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) ∧ 𝑔 ∈ dom 𝑆) → ¬ ((𝑍 ↾ dom 𝑆) ↾ suc 𝑔) <s ((𝑆 ↾ dom 𝑆) ↾ suc 𝑔))
140	139	ralrimiva 3157	. . . . 5 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) → ∀𝑔 ∈ dom 𝑆 ¬ ((𝑍 ↾ dom 𝑆) ↾ suc 𝑔) <s ((𝑆 ↾ dom 𝑆) ↾ suc 𝑔))
141		noresle 27861	. . . . 5 ⊢ ((((𝑆 ↾ dom 𝑆) ∈ No ∧ (𝑍 ↾ dom 𝑆) ∈ No ) ∧ (dom (𝑆 ↾ dom 𝑆) ⊆ dom 𝑆 ∧ dom (𝑍 ↾ dom 𝑆) ⊆ dom 𝑆 ∧ ∀𝑔 ∈ dom 𝑆 ¬ ((𝑍 ↾ dom 𝑆) ↾ suc 𝑔) <s ((𝑆 ↾ dom 𝑆) ↾ suc 𝑔))) → ¬ (𝑍 ↾ dom 𝑆) <s (𝑆 ↾ dom 𝑆))
142	72, 75, 79, 83, 140, 141	syl23anc 1404	. . . 4 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) → ¬ (𝑍 ↾ dom 𝑆) <s (𝑆 ↾ dom 𝑆))
143		nofun 27813	. . . . . 6 ⊢ (𝑆 ∈ No → Fun 𝑆)
144		funrel 6553	. . . . . 6 ⊢ (Fun 𝑆 → Rel 𝑆)
145		resdm 6025	. . . . . 6 ⊢ (Rel 𝑆 → (𝑆 ↾ dom 𝑆) = 𝑆)
146	68, 143, 144, 145	4syl 20	. . . . 5 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) → (𝑆 ↾ dom 𝑆) = 𝑆)
147	146	breq2d 5121	. . . 4 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) → ((𝑍 ↾ dom 𝑆) <s (𝑆 ↾ dom 𝑆) ↔ (𝑍 ↾ dom 𝑆) <s 𝑆))
148	142, 147	mtbid 327	. . 3 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) → ¬ (𝑍 ↾ dom 𝑆) <s 𝑆)
149	65, 148	pm2.61ian 823	. 2 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) → ¬ (𝑍 ↾ dom 𝑆) <s 𝑆)
150		simpll1 1231	. . . . . 6 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ¬ (𝑍 ↾ dom 𝑆) <s 𝑆) ∧ 𝑎 ∈ 𝐴) → 𝐴 ⊆ No )
151		simpll2 1232	. . . . . 6 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ¬ (𝑍 ↾ dom 𝑆) <s 𝑆) ∧ 𝑎 ∈ 𝐴) → 𝐴 ∈ V)
152		simpr 489	. . . . . 6 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ¬ (𝑍 ↾ dom 𝑆) <s 𝑆) ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ 𝐴)
153	3	nosupbnd1 27878	. . . . . 6 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑎 ∈ 𝐴) → (𝑎 ↾ dom 𝑆) <s 𝑆)
154	150, 151, 152, 153	syl3anc 1398	. . . . 5 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ¬ (𝑍 ↾ dom 𝑆) <s 𝑆) ∧ 𝑎 ∈ 𝐴) → (𝑎 ↾ dom 𝑆) <s 𝑆)
155		simplr 780	. . . . 5 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ¬ (𝑍 ↾ dom 𝑆) <s 𝑆) ∧ 𝑎 ∈ 𝐴) → ¬ (𝑍 ↾ dom 𝑆) <s 𝑆)
156		simpl1 1210	. . . . . . . 8 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ¬ (𝑍 ↾ dom 𝑆) <s 𝑆) → 𝐴 ⊆ No )
157	156	sselda 3937	. . . . . . 7 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ¬ (𝑍 ↾ dom 𝑆) <s 𝑆) ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ No )
158	150, 151, 66	syl2anc 595	. . . . . . . 8 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ¬ (𝑍 ↾ dom 𝑆) <s 𝑆) ∧ 𝑎 ∈ 𝐴) → 𝑆 ∈ No )
159	158, 69	syl 18	. . . . . . 7 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ¬ (𝑍 ↾ dom 𝑆) <s 𝑆) ∧ 𝑎 ∈ 𝐴) → dom 𝑆 ∈ On)
160		noreson 27824	. . . . . . 7 ⊢ ((𝑎 ∈ No ∧ dom 𝑆 ∈ On) → (𝑎 ↾ dom 𝑆) ∈ No )
161	157, 159, 160	syl2anc 595	. . . . . 6 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ¬ (𝑍 ↾ dom 𝑆) <s 𝑆) ∧ 𝑎 ∈ 𝐴) → (𝑎 ↾ dom 𝑆) ∈ No )
162		simpll3 1233	. . . . . . 7 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ¬ (𝑍 ↾ dom 𝑆) <s 𝑆) ∧ 𝑎 ∈ 𝐴) → 𝑍 ∈ No )
163	162, 159, 74	syl2anc 595	. . . . . 6 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ¬ (𝑍 ↾ dom 𝑆) <s 𝑆) ∧ 𝑎 ∈ 𝐴) → (𝑍 ↾ dom 𝑆) ∈ No )
164		sotr3 5610	. . . . . . 7 ⊢ (( <s Or No ∧ ((𝑎 ↾ dom 𝑆) ∈ No ∧ 𝑆 ∈ No ∧ (𝑍 ↾ dom 𝑆) ∈ No )) → (((𝑎 ↾ dom 𝑆) <s 𝑆 ∧ ¬ (𝑍 ↾ dom 𝑆) <s 𝑆) → (𝑎 ↾ dom 𝑆) <s (𝑍 ↾ dom 𝑆)))
165	96, 164	mpan 702	. . . . . 6 ⊢ (((𝑎 ↾ dom 𝑆) ∈ No ∧ 𝑆 ∈ No ∧ (𝑍 ↾ dom 𝑆) ∈ No ) → (((𝑎 ↾ dom 𝑆) <s 𝑆 ∧ ¬ (𝑍 ↾ dom 𝑆) <s 𝑆) → (𝑎 ↾ dom 𝑆) <s (𝑍 ↾ dom 𝑆)))
166	161, 158, 163, 165	syl3anc 1398	. . . . 5 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ¬ (𝑍 ↾ dom 𝑆) <s 𝑆) ∧ 𝑎 ∈ 𝐴) → (((𝑎 ↾ dom 𝑆) <s 𝑆 ∧ ¬ (𝑍 ↾ dom 𝑆) <s 𝑆) → (𝑎 ↾ dom 𝑆) <s (𝑍 ↾ dom 𝑆)))
167	154, 155, 166	mp2and 711	. . . 4 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ¬ (𝑍 ↾ dom 𝑆) <s 𝑆) ∧ 𝑎 ∈ 𝐴) → (𝑎 ↾ dom 𝑆) <s (𝑍 ↾ dom 𝑆))
168		ltsres 27826	. . . . 5 ⊢ ((𝑎 ∈ No ∧ 𝑍 ∈ No ∧ dom 𝑆 ∈ On) → ((𝑎 ↾ dom 𝑆) <s (𝑍 ↾ dom 𝑆) → 𝑎 <s 𝑍))
169	157, 162, 159, 168	syl3anc 1398	. . . 4 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ¬ (𝑍 ↾ dom 𝑆) <s 𝑆) ∧ 𝑎 ∈ 𝐴) → ((𝑎 ↾ dom 𝑆) <s (𝑍 ↾ dom 𝑆) → 𝑎 <s 𝑍))
170	167, 169	mpd 16	. . 3 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ¬ (𝑍 ↾ dom 𝑆) <s 𝑆) ∧ 𝑎 ∈ 𝐴) → 𝑎 <s 𝑍)
171	170	ralrimiva 3157	. 2 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ¬ (𝑍 ↾ dom 𝑆) <s 𝑆) → ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)
172	149, 171	impbida 812	1 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) → (∀𝑎 ∈ 𝐴 𝑎 <s 𝑍 ↔ ¬ (𝑍 ↾ dom 𝑆) <s 𝑆))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 {cab 2741 ∀wral 3079 ∃wrex 3089 ∃!wreu 3367 ∃*wrmo 3368 Vcvv 3455 ∪ cun 3903 ∩ cin 3904 ⊆ wss 3905 ifcif 4487 {csn 4589 ⟨cop 4595 class class class wbr 5109 ↦ cmpt 5192 Or wor 5568 dom cdm 5661 ↾ cres 5663 Rel wrel 5666 Ord word 6359 Oncon0 6360 suc csuc 6362 ℩cio 6490 Fun wfun 6530 ‘cfv 6536 ℩crio 7366 2_oc2o 8443 No csur 27804 <s clts 27805

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-int 4913 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-ord 6363 df-on 6364 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-1o 8449 df-2o 8450 df-no 27807 df-lts 27808 df-bday 27809

This theorem is referenced by: nosupinfsep 27896 noetasuplem4 27900

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