nosupbnd2 - Mathbox for Scott Fenton

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Theorem nosupbnd2 33919

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 𝑆,𝑎,𝑔 𝑣,𝑢,𝑥,𝑦 𝑍,𝑎,𝑔,𝑥 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 1917	. . . . . 6 ⊢ Ⅎ𝑥((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)
2		nfcv 2907	. . . . . . . . 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 3239	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦
5		nfriota1 7239	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
6	5	nfdm 5860	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
7		nfcv 2907	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥2_o
8	6, 7	nfop 4820	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2_o⟩
9	8	nfsn 4643	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥{⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2_o⟩}
10	5, 9	nfun 4099	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2_o⟩})
11		nfcv 2907	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥{𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}
12		nfiota1 6393	. . . . . . . . . . . . 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 5181	. . . . . . . . . . . 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 4489	. . . . . . . . . . 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 2905	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑆
16	15	nfdm 5860	. . . . . . . . 9 ⊢ Ⅎ𝑥dom 𝑆
17	2, 16	nfres 5893	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑍 ↾ dom 𝑆)
18		nfcv 2907	. . . . . . . 8 ⊢ Ⅎ𝑥 <s
19	17, 18, 15	nfbr 5121	. . . . . . 7 ⊢ Ⅎ𝑥(𝑍 ↾ dom 𝑆) <s 𝑆
20	19	nfn 1860	. . . . . 6 ⊢ Ⅎ𝑥 ¬ (𝑍 ↾ dom 𝑆) <s 𝑆
21	1, 20	nfim 1899	. . . . 5 ⊢ Ⅎ𝑥(((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) → ¬ (𝑍 ↾ dom 𝑆) <s 𝑆)
22		simpl 483	. . . . . . . . 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 3237	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
24	23	adantr 481	. . . . . . . . . . 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 33901	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ No → ∃*𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
26	25	3ad2ant1 1132	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) → ∃*𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
27	26	ad2antrl 725	. . . . . . . . . . 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 3361	. . . . . . . . . . 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 583	. . . . . . . . . 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 7254	. . . . . . . . . 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 17	. . . . . . . . 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 231	. . . . . . . 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 33918	. . . . . . . . . 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 1119	. . . . . . . . 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 5812	. . . . . . . . . . . . 13 ⊢ ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) = 𝑥 → dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) = dom 𝑥)
36		suceq 6331	. . . . . . . . . . . . 13 ⊢ (dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) = dom 𝑥 → suc dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) = suc dom 𝑥)
37	35, 36	syl 17	. . . . . . . . . . . 12 ⊢ ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) = 𝑥 → suc dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) = suc dom 𝑥)
38	37	reseq2d 5891	. . . . . . . . . . 11 ⊢ ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) = 𝑥 → (𝑍 ↾ suc dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)) = (𝑍 ↾ suc dom 𝑥))
39		id 22	. . . . . . . . . . . 12 ⊢ ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) = 𝑥 → (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) = 𝑥)
40	35	opeq1d 4810	. . . . . . . . . . . . 13 ⊢ ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) = 𝑥 → ⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2_o⟩ = ⟨dom 𝑥, 2_o⟩)
41	40	sneqd 4573	. . . . . . . . . . . 12 ⊢ ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) = 𝑥 → {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2_o⟩} = {⟨dom 𝑥, 2_o⟩})
42	39, 41	uneq12d 4098	. . . . . . . . . . 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⟩}))
43	38, 42	breq12d 5087	. . . . . . . . . 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⟩})))
44	43	notbid 318	. . . . . . . . 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⟩})))
45	34, 44	syl5ibrcom 246	. . . . . . . 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⟩})))
46	32, 45	mpd 15	. . . . . . 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⟩}))
47		iftrue 4465	. . . . . . . . . . . . . 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⟩}))
48	3, 47	eqtrid 2790	. . . . . . . . . . . . 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⟩}))
49	23, 48	syl 17	. . . . . . . . . . . 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⟩}))
50	49	dmeqd 5814	. . . . . . . . . . 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⟩}))
51		2on 8311	. . . . . . . . . . . . . . 15 ⊢ 2_o ∈ On
52	51	elexi 3451	. . . . . . . . . . . . . 14 ⊢ 2_o ∈ V
53	52	dmsnop 6119	. . . . . . . . . . . . 13 ⊢ dom {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2_o⟩} = {dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)}
54	53	uneq2i 4094	. . . . . . . . . . . 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)})
55		dmun 5819	. . . . . . . . . . . 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⟩})
56		df-suc 6272	. . . . . . . . . . . 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)})
57	54, 55, 56	3eqtr4i 2776	. . . . . . . . . . 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)
58	50, 57	eqtrdi 2794	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) → dom 𝑆 = suc dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦))
59	58	reseq2d 5891	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) → (𝑍 ↾ dom 𝑆) = (𝑍 ↾ suc dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)))
60	59	adantr 481	. . . . . . . 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)))
61	49	adantr 481	. . . . . . . 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⟩}))
62	60, 61	breq12d 5087	. . . . . . 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⟩})))
63	46, 62	mtbird 325	. . . . . 6 ⊢ (((𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) → ¬ (𝑍 ↾ dom 𝑆) <s 𝑆)
64	63	exp31 420	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 → (((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) → ¬ (𝑍 ↾ dom 𝑆) <s 𝑆)))
65	21, 64	rexlimi 3248	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 → (((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) → ¬ (𝑍 ↾ dom 𝑆) <s 𝑆))
66	65	imp 407	. . 3 ⊢ ((∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) → ¬ (𝑍 ↾ dom 𝑆) <s 𝑆)
67	3	nosupno 33906	. . . . . . . 8 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V) → 𝑆 ∈ No )
68	67	3adant3 1131	. . . . . . 7 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) → 𝑆 ∈ No )
69	68	ad2antrl 725	. . . . . 6 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) → 𝑆 ∈ No )
70		nodmon 33853	. . . . . . 7 ⊢ (𝑆 ∈ No → dom 𝑆 ∈ On)
71	69, 70	syl 17	. . . . . 6 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) → dom 𝑆 ∈ On)
72		noreson 33863	. . . . . 6 ⊢ ((𝑆 ∈ No ∧ dom 𝑆 ∈ On) → (𝑆 ↾ dom 𝑆) ∈ No )
73	69, 71, 72	syl2anc 584	. . . . 5 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) → (𝑆 ↾ dom 𝑆) ∈ No )
74		simprl3 1219	. . . . . 6 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) → 𝑍 ∈ No )
75		noreson 33863	. . . . . 6 ⊢ ((𝑍 ∈ No ∧ dom 𝑆 ∈ On) → (𝑍 ↾ dom 𝑆) ∈ No )
76	74, 71, 75	syl2anc 584	. . . . 5 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) → (𝑍 ↾ dom 𝑆) ∈ No )
77		dmres 5913	. . . . . . 7 ⊢ dom (𝑆 ↾ dom 𝑆) = (dom 𝑆 ∩ dom 𝑆)
78		inss2 4163	. . . . . . 7 ⊢ (dom 𝑆 ∩ dom 𝑆) ⊆ dom 𝑆
79	77, 78	eqsstri 3955	. . . . . 6 ⊢ dom (𝑆 ↾ dom 𝑆) ⊆ dom 𝑆
80	79	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)
81		dmres 5913	. . . . . . 7 ⊢ dom (𝑍 ↾ dom 𝑆) = (dom 𝑆 ∩ dom 𝑍)
82		inss1 4162	. . . . . . 7 ⊢ (dom 𝑆 ∩ dom 𝑍) ⊆ dom 𝑆
83	81, 82	eqsstri 3955	. . . . . 6 ⊢ dom (𝑍 ↾ dom 𝑆) ⊆ dom 𝑆
84	83	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)
85	3	nosupdm 33907	. . . . . . . . . . 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)))})
86	85	abeq2d 2874	. . . . . . . . . 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)))))
87	86	adantr 481	. . . . . . . . 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)))))
88		simprl 768	. . . . . . . . . . . . . 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)
89		simplrr 775	. . . . . . . . . . . . . 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)
90		breq1 5077	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑝 → (𝑎 <s 𝑍 ↔ 𝑝 <s 𝑍))
91	90	rspcv 3557	. . . . . . . . . . . . . 14 ⊢ (𝑝 ∈ 𝐴 → (∀𝑎 ∈ 𝐴 𝑎 <s 𝑍 → 𝑝 <s 𝑍))
92	88, 89, 91	sylc 65	. . . . . . . . . . . . 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)
93		simprl1 1217	. . . . . . . . . . . . . . . 16 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) → 𝐴 ⊆ No )
94	93	adantr 481	. . . . . . . . . . . . . . 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 )
95	94, 88	sseldd 3922	. . . . . . . . . . . . . 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 )
96	74	adantr 481	. . . . . . . . . . . . . 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 )
97		sltso 33879	. . . . . . . . . . . . . . 15 ⊢ <s Or No
98		soasym 5534	. . . . . . . . . . . . . . 15 ⊢ (( <s Or No ∧ (𝑝 ∈ No ∧ 𝑍 ∈ No )) → (𝑝 <s 𝑍 → ¬ 𝑍 <s 𝑝))
99	97, 98	mpan 687	. . . . . . . . . . . . . 14 ⊢ ((𝑝 ∈ No ∧ 𝑍 ∈ No ) → (𝑝 <s 𝑍 → ¬ 𝑍 <s 𝑝))
100	95, 96, 99	syl2anc 584	. . . . . . . . . . . . 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))
101	92, 100	mpd 15	. . . . . . . . . . . 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)
102		nodmon 33853	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 ∈ No → dom 𝑝 ∈ On)
103	95, 102	syl 17	. . . . . . . . . . . . . . 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)
104		simprrl 778	. . . . . . . . . . . . . . 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)
105		onelon 6291	. . . . . . . . . . . . . . 15 ⊢ ((dom 𝑝 ∈ On ∧ 𝑔 ∈ dom 𝑝) → 𝑔 ∈ On)
106	103, 104, 105	syl2anc 584	. . . . . . . . . . . . . 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)
107		sucelon 7664	. . . . . . . . . . . . . 14 ⊢ (𝑔 ∈ On ↔ suc 𝑔 ∈ On)
108	106, 107	sylib 217	. . . . . . . . . . . . 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)
109		sltres 33865	. . . . . . . . . . . . 13 ⊢ ((𝑍 ∈ No ∧ 𝑝 ∈ No ∧ suc 𝑔 ∈ On) → ((𝑍 ↾ suc 𝑔) <s (𝑝 ↾ suc 𝑔) → 𝑍 <s 𝑝))
110	96, 95, 108, 109	syl3anc 1370	. . . . . . . . . . . 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))
111	101, 110	mtod 197	. . . . . . . . . . 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))
112		simpll 764	. . . . . . . . . . . . 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)
113		simprl2 1218	. . . . . . . . . . . . . . 15 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) → 𝐴 ∈ V)
114	93, 113	jca 512	. . . . . . . . . . . . . 14 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) → (𝐴 ⊆ No ∧ 𝐴 ∈ V))
115	114	adantr 481	. . . . . . . . . . . . 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))
116		simprrr 779	. . . . . . . . . . . . . 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)))
117		breq1 5077	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = 𝑞 → (𝑣 <s 𝑝 ↔ 𝑞 <s 𝑝))
118	117	notbid 318	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 𝑞 → (¬ 𝑣 <s 𝑝 ↔ ¬ 𝑞 <s 𝑝))
119		reseq1 5885	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = 𝑞 → (𝑣 ↾ suc 𝑔) = (𝑞 ↾ suc 𝑔))
120	119	eqeq2d 2749	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 𝑞 → ((𝑝 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔) ↔ (𝑝 ↾ suc 𝑔) = (𝑞 ↾ suc 𝑔)))
121	118, 120	imbi12d 345	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = 𝑞 → ((¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ↔ (¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑔) = (𝑞 ↾ suc 𝑔))))
122	121	cbvralvw 3383	. . . . . . . . . . . . . 14 ⊢ (∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑝 → (𝑝 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ↔ ∀𝑞 ∈ 𝐴 (¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑔) = (𝑞 ↾ suc 𝑔)))
123	116, 122	sylibr 233	. . . . . . . . . . . . 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)))
124	3	nosupres 33910	. . . . . . . . . . . . 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))
125	112, 115, 88, 104, 123, 124	syl113anc 1381	. . . . . . . . . . . 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))
126	125	breq2d 5086	. . . . . . . . . . 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)))
127	111, 126	mtbird 325	. . . . . . . . . 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))
128	127	rexlimdvaa 3214	. . . . . . . . 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)))
129	87, 128	sylbid 239	. . . . . . . 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)))
130	129	imp 407	. . . . . . 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))
131	69	adantr 481	. . . . . . . . . . 11 ⊢ (((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) ∧ 𝑔 ∈ dom 𝑆) → 𝑆 ∈ No )
132		nodmord 33856	. . . . . . . . . . 11 ⊢ (𝑆 ∈ No → Ord dom 𝑆)
133	131, 132	syl 17	. . . . . . . . . 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)
134		simpr 485	. . . . . . . . . 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)
135		ordsucss 7665	. . . . . . . . . 10 ⊢ (Ord dom 𝑆 → (𝑔 ∈ dom 𝑆 → suc 𝑔 ⊆ dom 𝑆))
136	133, 134, 135	sylc 65	. . . . . . . . 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)
137	136	resabs1d 5922	. . . . . . . 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))
138	136	resabs1d 5922	. . . . . . . 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))
139	137, 138	breq12d 5087	. . . . . . 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)))
140	130, 139	mtbird 325	. . . . . 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))
141	140	ralrimiva 3103	. . . . 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))
142		noresle 33900	. . . . 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))
143	73, 76, 80, 84, 141, 142	syl23anc 1376	. . . 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))
144		nofun 33852	. . . . . . 7 ⊢ (𝑆 ∈ No → Fun 𝑆)
145	69, 144	syl 17	. . . . . 6 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) → Fun 𝑆)
146		funrel 6451	. . . . . 6 ⊢ (Fun 𝑆 → Rel 𝑆)
147		resdm 5936	. . . . . 6 ⊢ (Rel 𝑆 → (𝑆 ↾ dom 𝑆) = 𝑆)
148	145, 146, 147	3syl 18	. . . . 5 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) → (𝑆 ↾ dom 𝑆) = 𝑆)
149	148	breq2d 5086	. . . 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))
150	143, 149	mtbid 324	. . 3 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)) → ¬ (𝑍 ↾ dom 𝑆) <s 𝑆)
151	66, 150	pm2.61ian 809	. 2 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) → ¬ (𝑍 ↾ dom 𝑆) <s 𝑆)
152		simpll1 1211	. . . . . 6 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ¬ (𝑍 ↾ dom 𝑆) <s 𝑆) ∧ 𝑎 ∈ 𝐴) → 𝐴 ⊆ No )
153		simpll2 1212	. . . . . 6 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ¬ (𝑍 ↾ dom 𝑆) <s 𝑆) ∧ 𝑎 ∈ 𝐴) → 𝐴 ∈ V)
154		simpr 485	. . . . . 6 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ¬ (𝑍 ↾ dom 𝑆) <s 𝑆) ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ 𝐴)
155	3	nosupbnd1 33917	. . . . . 6 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑎 ∈ 𝐴) → (𝑎 ↾ dom 𝑆) <s 𝑆)
156	152, 153, 154, 155	syl3anc 1370	. . . . 5 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ¬ (𝑍 ↾ dom 𝑆) <s 𝑆) ∧ 𝑎 ∈ 𝐴) → (𝑎 ↾ dom 𝑆) <s 𝑆)
157		simplr 766	. . . . 5 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ¬ (𝑍 ↾ dom 𝑆) <s 𝑆) ∧ 𝑎 ∈ 𝐴) → ¬ (𝑍 ↾ dom 𝑆) <s 𝑆)
158		simpl1 1190	. . . . . . . 8 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ¬ (𝑍 ↾ dom 𝑆) <s 𝑆) → 𝐴 ⊆ No )
159	158	sselda 3921	. . . . . . 7 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ¬ (𝑍 ↾ dom 𝑆) <s 𝑆) ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ No )
160	152, 153, 67	syl2anc 584	. . . . . . . 8 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ¬ (𝑍 ↾ dom 𝑆) <s 𝑆) ∧ 𝑎 ∈ 𝐴) → 𝑆 ∈ No )
161	160, 70	syl 17	. . . . . . 7 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ¬ (𝑍 ↾ dom 𝑆) <s 𝑆) ∧ 𝑎 ∈ 𝐴) → dom 𝑆 ∈ On)
162		noreson 33863	. . . . . . 7 ⊢ ((𝑎 ∈ No ∧ dom 𝑆 ∈ On) → (𝑎 ↾ dom 𝑆) ∈ No )
163	159, 161, 162	syl2anc 584	. . . . . 6 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ¬ (𝑍 ↾ dom 𝑆) <s 𝑆) ∧ 𝑎 ∈ 𝐴) → (𝑎 ↾ dom 𝑆) ∈ No )
164		simpll3 1213	. . . . . . 7 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ¬ (𝑍 ↾ dom 𝑆) <s 𝑆) ∧ 𝑎 ∈ 𝐴) → 𝑍 ∈ No )
165	164, 161, 75	syl2anc 584	. . . . . 6 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ¬ (𝑍 ↾ dom 𝑆) <s 𝑆) ∧ 𝑎 ∈ 𝐴) → (𝑍 ↾ dom 𝑆) ∈ No )
166		sotr3 33733	. . . . . . 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)))
167	97, 166	mpan 687	. . . . . 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)))
168	163, 160, 165, 167	syl3anc 1370	. . . . 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)))
169	156, 157, 168	mp2and 696	. . . 4 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ¬ (𝑍 ↾ dom 𝑆) <s 𝑆) ∧ 𝑎 ∈ 𝐴) → (𝑎 ↾ dom 𝑆) <s (𝑍 ↾ dom 𝑆))
170		sltres 33865	. . . . 5 ⊢ ((𝑎 ∈ No ∧ 𝑍 ∈ No ∧ dom 𝑆 ∈ On) → ((𝑎 ↾ dom 𝑆) <s (𝑍 ↾ dom 𝑆) → 𝑎 <s 𝑍))
171	159, 164, 161, 170	syl3anc 1370	. . . 4 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ¬ (𝑍 ↾ dom 𝑆) <s 𝑆) ∧ 𝑎 ∈ 𝐴) → ((𝑎 ↾ dom 𝑆) <s (𝑍 ↾ dom 𝑆) → 𝑎 <s 𝑍))
172	169, 171	mpd 15	. . 3 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ¬ (𝑍 ↾ dom 𝑆) <s 𝑆) ∧ 𝑎 ∈ 𝐴) → 𝑎 <s 𝑍)
173	172	ralrimiva 3103	. 2 ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ¬ (𝑍 ↾ dom 𝑆) <s 𝑆) → ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)
174	151, 173	impbida 798	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 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 {cab 2715 ∀wral 3064 ∃wrex 3065 ∃!wreu 3066 ∃*wrmo 3067 Vcvv 3432 ∪ cun 3885 ∩ cin 3886 ⊆ wss 3887 ifcif 4459 {csn 4561 ⟨cop 4567 class class class wbr 5074 ↦ cmpt 5157 Or wor 5502 dom cdm 5589 ↾ cres 5591 Rel wrel 5594 Ord word 6265 Oncon0 6266 suc csuc 6268 ℩cio 6389 Fun wfun 6427 ‘cfv 6433 ℩crio 7231 2_oc2o 8291 No csur 33843 <s cslt 33844

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pr 5352 ax-un 7588

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-ord 6269 df-on 6270 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-1o 8297 df-2o 8298 df-no 33846 df-slt 33847 df-bday 33848

This theorem is referenced by: nosupinfsep 33935 noetasuplem4 33939

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