noinfno - Mathbox for Scott Fenton

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Theorem noinfno 33510

Description: The next several theorems deal with a surreal "infimum". This surreal will ultimately be shown to bound 𝐵 above and bound the restriction of any surreal below. We begin by showing that the given expression actually defines a surreal number. (Contributed by Scott Fenton, 8-Aug-2024.)

**Hypothesis**
Ref	Expression
noinfno.1	⊢ 𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1_o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))

**Assertion**
Ref	Expression
noinfno	⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) → 𝑇 ∈ No )

Distinct variable groups: 𝑥,𝐵,𝑦,𝑔,𝑣,𝑢 𝑔,𝑉 Allowed substitution hints: 𝑇(𝑥,𝑦,𝑣,𝑢,𝑔) 𝑉(𝑥,𝑦,𝑣,𝑢)

Proof of Theorem noinfno Dummy variables 𝑎 𝑏 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		noinfno.1	. 2 ⊢ 𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1_o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))
2		iftrue 4429	. . . . 5 ⊢ (∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 → if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1_o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) = ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1_o⟩}))
3	2	adantr 484	. . . 4 ⊢ ((∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉)) → if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1_o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) = ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1_o⟩}))
4		simprl 770	. . . . . 6 ⊢ ((∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉)) → 𝐵 ⊆ No )
5		simpl 486	. . . . . . . 8 ⊢ ((∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉)) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥)
6		nominmo 33491	. . . . . . . . 9 ⊢ (𝐵 ⊆ No → ∃*𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥)
7	6	ad2antrl 727	. . . . . . . 8 ⊢ ((∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉)) → ∃*𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥)
8		reu5 3340	. . . . . . . 8 ⊢ (∃!𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ↔ (∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ∃*𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥))
9	5, 7, 8	sylanbrc 586	. . . . . . 7 ⊢ ((∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉)) → ∃!𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥)
10		riotacl 7130	. . . . . . 7 ⊢ (∃!𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 → (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∈ 𝐵)
11	9, 10	syl 17	. . . . . 6 ⊢ ((∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉)) → (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∈ 𝐵)
12	4, 11	sseldd 3895	. . . . 5 ⊢ ((∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉)) → (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∈ No )
13		1oex 8125	. . . . . . 7 ⊢ 1_o ∈ V
14	13	prid1 4658	. . . . . 6 ⊢ 1_o ∈ {1_o, 2_o}
15	14	noextend 33458	. . . . 5 ⊢ ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∈ No → ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1_o⟩}) ∈ No )
16	12, 15	syl 17	. . . 4 ⊢ ((∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉)) → ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1_o⟩}) ∈ No )
17	3, 16	eqeltrd 2852	. . 3 ⊢ ((∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉)) → if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1_o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) ∈ No )
18		iffalse 4432	. . . . 5 ⊢ (¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 → if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1_o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) = (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))
19	18	adantr 484	. . . 4 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉)) → if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1_o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) = (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))
20		funmpt 6377	. . . . . 6 ⊢ Fun (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))
21	20	a1i 11	. . . . 5 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉)) → Fun (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))
22		iotaex 6319	. . . . . . 7 ⊢ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)) ∈ V
23		eqid 2758	. . . . . . 7 ⊢ (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))) = (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))
24	22, 23	dmmpti 6479	. . . . . 6 ⊢ dom (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))) = {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}
25		simpl 486	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ⊆ No ∧ (𝑢 ∈ 𝐵 ∧ (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))))) → 𝐵 ⊆ No )
26		simprl 770	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ⊆ No ∧ (𝑢 ∈ 𝐵 ∧ (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))))) → 𝑢 ∈ 𝐵)
27	25, 26	sseldd 3895	. . . . . . . . . . . . 13 ⊢ ((𝐵 ⊆ No ∧ (𝑢 ∈ 𝐵 ∧ (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))))) → 𝑢 ∈ No )
28		nodmon 33442	. . . . . . . . . . . . 13 ⊢ (𝑢 ∈ No → dom 𝑢 ∈ On)
29	27, 28	syl 17	. . . . . . . . . . . 12 ⊢ ((𝐵 ⊆ No ∧ (𝑢 ∈ 𝐵 ∧ (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))))) → dom 𝑢 ∈ On)
30		simprrl 780	. . . . . . . . . . . 12 ⊢ ((𝐵 ⊆ No ∧ (𝑢 ∈ 𝐵 ∧ (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))))) → 𝑦 ∈ dom 𝑢)
31		onelon 6198	. . . . . . . . . . . 12 ⊢ ((dom 𝑢 ∈ On ∧ 𝑦 ∈ dom 𝑢) → 𝑦 ∈ On)
32	29, 30, 31	syl2anc 587	. . . . . . . . . . 11 ⊢ ((𝐵 ⊆ No ∧ (𝑢 ∈ 𝐵 ∧ (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))))) → 𝑦 ∈ On)
33	32	rexlimdvaa 3209	. . . . . . . . . 10 ⊢ (𝐵 ⊆ No → (∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) → 𝑦 ∈ On))
34	33	abssdv 3975	. . . . . . . . 9 ⊢ (𝐵 ⊆ No → {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ⊆ On)
35		simplr 768	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐵 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐵) → 𝑎 ∈ 𝑏)
36		ssel2 3889	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 ⊆ No ∧ 𝑢 ∈ 𝐵) → 𝑢 ∈ No )
37	36	adantlr 714	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐵 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐵) → 𝑢 ∈ No )
38	37, 28	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐵 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐵) → dom 𝑢 ∈ On)
39		ontr1 6219	. . . . . . . . . . . . . . . . . 18 ⊢ (dom 𝑢 ∈ On → ((𝑎 ∈ 𝑏 ∧ 𝑏 ∈ dom 𝑢) → 𝑎 ∈ dom 𝑢))
40	38, 39	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐵 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐵) → ((𝑎 ∈ 𝑏 ∧ 𝑏 ∈ dom 𝑢) → 𝑎 ∈ dom 𝑢))
41	35, 40	mpand 694	. . . . . . . . . . . . . . . 16 ⊢ (((𝐵 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐵) → (𝑏 ∈ dom 𝑢 → 𝑎 ∈ dom 𝑢))
42	41	adantrd 495	. . . . . . . . . . . . . . 15 ⊢ (((𝐵 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐵) → ((𝑏 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏))) → 𝑎 ∈ dom 𝑢))
43		reseq1 5821	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏) → ((𝑢 ↾ suc 𝑏) ↾ suc 𝑎) = ((𝑣 ↾ suc 𝑏) ↾ suc 𝑎))
44		onelon 6198	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((dom 𝑢 ∈ On ∧ 𝑏 ∈ dom 𝑢) → 𝑏 ∈ On)
45	38, 44	sylan 583	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐵 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐵) ∧ 𝑏 ∈ dom 𝑢) → 𝑏 ∈ On)
46		sucelon 7536	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑏 ∈ On ↔ suc 𝑏 ∈ On)
47	45, 46	sylib 221	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐵 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐵) ∧ 𝑏 ∈ dom 𝑢) → suc 𝑏 ∈ On)
48		simpllr 775	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐵 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐵) ∧ 𝑏 ∈ dom 𝑢) → 𝑎 ∈ 𝑏)
49		eloni 6183	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑏 ∈ On → Ord 𝑏)
50	45, 49	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐵 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐵) ∧ 𝑏 ∈ dom 𝑢) → Ord 𝑏)
51		ordsucelsuc 7541	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (Ord 𝑏 → (𝑎 ∈ 𝑏 ↔ suc 𝑎 ∈ suc 𝑏))
52	50, 51	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐵 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐵) ∧ 𝑏 ∈ dom 𝑢) → (𝑎 ∈ 𝑏 ↔ suc 𝑎 ∈ suc 𝑏))
53	48, 52	mpbid 235	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐵 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐵) ∧ 𝑏 ∈ dom 𝑢) → suc 𝑎 ∈ suc 𝑏)
54		onelss 6215	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (suc 𝑏 ∈ On → (suc 𝑎 ∈ suc 𝑏 → suc 𝑎 ⊆ suc 𝑏))
55	47, 53, 54	sylc 65	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐵 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐵) ∧ 𝑏 ∈ dom 𝑢) → suc 𝑎 ⊆ suc 𝑏)
56	55	resabs1d 5858	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐵 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐵) ∧ 𝑏 ∈ dom 𝑢) → ((𝑢 ↾ suc 𝑏) ↾ suc 𝑎) = (𝑢 ↾ suc 𝑎))
57	55	resabs1d 5858	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐵 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐵) ∧ 𝑏 ∈ dom 𝑢) → ((𝑣 ↾ suc 𝑏) ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎))
58	56, 57	eqeq12d 2774	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐵 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐵) ∧ 𝑏 ∈ dom 𝑢) → (((𝑢 ↾ suc 𝑏) ↾ suc 𝑎) = ((𝑣 ↾ suc 𝑏) ↾ suc 𝑎) ↔ (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎)))
59	43, 58	syl5ib 247	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐵 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐵) ∧ 𝑏 ∈ dom 𝑢) → ((𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏) → (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎)))
60	59	imim2d 57	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐵 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐵) ∧ 𝑏 ∈ dom 𝑢) → ((¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏)) → (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎))))
61	60	ralimdv 3109	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐵 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐵) ∧ 𝑏 ∈ dom 𝑢) → (∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏)) → ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎))))
62	61	expimpd 457	. . . . . . . . . . . . . . 15 ⊢ (((𝐵 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐵) → ((𝑏 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏))) → ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎))))
63	42, 62	jcad 516	. . . . . . . . . . . . . 14 ⊢ (((𝐵 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐵) → ((𝑏 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏))) → (𝑎 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎)))))
64	63	reximdva 3198	. . . . . . . . . . . . 13 ⊢ ((𝐵 ⊆ No ∧ 𝑎 ∈ 𝑏) → (∃𝑢 ∈ 𝐵 (𝑏 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏))) → ∃𝑢 ∈ 𝐵 (𝑎 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎)))))
65	64	expimpd 457	. . . . . . . . . . . 12 ⊢ (𝐵 ⊆ No → ((𝑎 ∈ 𝑏 ∧ ∃𝑢 ∈ 𝐵 (𝑏 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏)))) → ∃𝑢 ∈ 𝐵 (𝑎 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎)))))
66		vex 3413	. . . . . . . . . . . . . 14 ⊢ 𝑏 ∈ V
67		eleq1w 2834	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑏 → (𝑦 ∈ dom 𝑢 ↔ 𝑏 ∈ dom 𝑢))
68		suceq 6238	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑏 → suc 𝑦 = suc 𝑏)
69	68	reseq2d 5827	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑏 → (𝑢 ↾ suc 𝑦) = (𝑢 ↾ suc 𝑏))
70	68	reseq2d 5827	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑏 → (𝑣 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑏))
71	69, 70	eqeq12d 2774	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑏 → ((𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦) ↔ (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏)))
72	71	imbi2d 344	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑏 → ((¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏))))
73	72	ralbidv 3126	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑏 → (∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏))))
74	67, 73	anbi12d 633	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑏 → ((𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) ↔ (𝑏 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏)))))
75	74	rexbidv 3221	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑏 → (∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) ↔ ∃𝑢 ∈ 𝐵 (𝑏 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏)))))
76	66, 75	elab 3590	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↔ ∃𝑢 ∈ 𝐵 (𝑏 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏))))
77	76	anbi2i 625	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ 𝑏 ∧ 𝑏 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}) ↔ (𝑎 ∈ 𝑏 ∧ ∃𝑢 ∈ 𝐵 (𝑏 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏)))))
78		vex 3413	. . . . . . . . . . . . 13 ⊢ 𝑎 ∈ V
79		eleq1w 2834	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑎 → (𝑦 ∈ dom 𝑢 ↔ 𝑎 ∈ dom 𝑢))
80		suceq 6238	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑎 → suc 𝑦 = suc 𝑎)
81	80	reseq2d 5827	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑎 → (𝑢 ↾ suc 𝑦) = (𝑢 ↾ suc 𝑎))
82	80	reseq2d 5827	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑎 → (𝑣 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑎))
83	81, 82	eqeq12d 2774	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑎 → ((𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦) ↔ (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎)))
84	83	imbi2d 344	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑎 → ((¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎))))
85	84	ralbidv 3126	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑎 → (∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎))))
86	79, 85	anbi12d 633	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑎 → ((𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) ↔ (𝑎 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎)))))
87	86	rexbidv 3221	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑎 → (∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) ↔ ∃𝑢 ∈ 𝐵 (𝑎 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎)))))
88	78, 87	elab 3590	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↔ ∃𝑢 ∈ 𝐵 (𝑎 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎))))
89	65, 77, 88	3imtr4g 299	. . . . . . . . . . 11 ⊢ (𝐵 ⊆ No → ((𝑎 ∈ 𝑏 ∧ 𝑏 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}) → 𝑎 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}))
90	89	alrimivv 1929	. . . . . . . . . 10 ⊢ (𝐵 ⊆ No → ∀𝑎∀𝑏((𝑎 ∈ 𝑏 ∧ 𝑏 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}) → 𝑎 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}))
91		dftr2 5143	. . . . . . . . . 10 ⊢ (Tr {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↔ ∀𝑎∀𝑏((𝑎 ∈ 𝑏 ∧ 𝑏 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}) → 𝑎 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}))
92	90, 91	sylibr 237	. . . . . . . . 9 ⊢ (𝐵 ⊆ No → Tr {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))})
93		dford5 33193	. . . . . . . . 9 ⊢ (Ord {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↔ ({𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ⊆ On ∧ Tr {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}))
94	34, 92, 93	sylanbrc 586	. . . . . . . 8 ⊢ (𝐵 ⊆ No → Ord {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))})
95	94	ad2antrl 727	. . . . . . 7 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉)) → Ord {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))})
96		bdayfo 33469	. . . . . . . . . . . . . 14 ⊢ bday : No –onto→On
97		fofun 6581	. . . . . . . . . . . . . 14 ⊢ ( bday : No –onto→On → Fun bday )
98	96, 97	ax-mp 5	. . . . . . . . . . . . 13 ⊢ Fun bday
99		funimaexg 6425	. . . . . . . . . . . . 13 ⊢ ((Fun bday ∧ 𝐵 ∈ 𝑉) → ( bday “ 𝐵) ∈ V)
100	98, 99	mpan 689	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ 𝑉 → ( bday “ 𝐵) ∈ V)
101	100	uniexd 7471	. . . . . . . . . . 11 ⊢ (𝐵 ∈ 𝑉 → ∪ ( bday “ 𝐵) ∈ V)
102	101	adantl 485	. . . . . . . . . 10 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) → ∪ ( bday “ 𝐵) ∈ V)
103	102	adantl 485	. . . . . . . . 9 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉)) → ∪ ( bday “ 𝐵) ∈ V)
104		bdayval 33440	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ No → ( bday ‘𝑢) = dom 𝑢)
105	27, 104	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ⊆ No ∧ (𝑢 ∈ 𝐵 ∧ (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))))) → ( bday ‘𝑢) = dom 𝑢)
106		fofn 6582	. . . . . . . . . . . . . . . . . 18 ⊢ ( bday : No –onto→On → bday Fn No )
107	96, 106	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ bday Fn No
108		fnfvima 6992	. . . . . . . . . . . . . . . . 17 ⊢ (( bday Fn No ∧ 𝐵 ⊆ No ∧ 𝑢 ∈ 𝐵) → ( bday ‘𝑢) ∈ ( bday “ 𝐵))
109	107, 108	mp3an1 1445	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ⊆ No ∧ 𝑢 ∈ 𝐵) → ( bday ‘𝑢) ∈ ( bday “ 𝐵))
110	109	adantrr 716	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ⊆ No ∧ (𝑢 ∈ 𝐵 ∧ (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))))) → ( bday ‘𝑢) ∈ ( bday “ 𝐵))
111	105, 110	eqeltrrd 2853	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ⊆ No ∧ (𝑢 ∈ 𝐵 ∧ (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))))) → dom 𝑢 ∈ ( bday “ 𝐵))
112		elssuni 4833	. . . . . . . . . . . . . 14 ⊢ (dom 𝑢 ∈ ( bday “ 𝐵) → dom 𝑢 ⊆ ∪ ( bday “ 𝐵))
113	111, 112	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝐵 ⊆ No ∧ (𝑢 ∈ 𝐵 ∧ (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))))) → dom 𝑢 ⊆ ∪ ( bday “ 𝐵))
114	113, 30	sseldd 3895	. . . . . . . . . . . 12 ⊢ ((𝐵 ⊆ No ∧ (𝑢 ∈ 𝐵 ∧ (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))))) → 𝑦 ∈ ∪ ( bday “ 𝐵))
115	114	rexlimdvaa 3209	. . . . . . . . . . 11 ⊢ (𝐵 ⊆ No → (∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) → 𝑦 ∈ ∪ ( bday “ 𝐵)))
116	115	abssdv 3975	. . . . . . . . . 10 ⊢ (𝐵 ⊆ No → {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ⊆ ∪ ( bday “ 𝐵))
117	116	ad2antrl 727	. . . . . . . . 9 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉)) → {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ⊆ ∪ ( bday “ 𝐵))
118	103, 117	ssexd 5197	. . . . . . . 8 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉)) → {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ∈ V)
119		elong 6181	. . . . . . . 8 ⊢ ({𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ∈ V → ({𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ∈ On ↔ Ord {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}))
120	118, 119	syl 17	. . . . . . 7 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉)) → ({𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ∈ On ↔ Ord {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}))
121	95, 120	mpbird 260	. . . . . 6 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉)) → {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ∈ On)
122	24, 121	eqeltrid 2856	. . . . 5 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉)) → dom (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))) ∈ On)
123	23	rnmpt 5800	. . . . . 6 ⊢ ran (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))) = {𝑧 ∣ ∃𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}𝑧 = (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))}
124		eleq1w 2834	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑔 → (𝑦 ∈ dom 𝑢 ↔ 𝑔 ∈ dom 𝑢))
125		suceq 6238	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑔 → suc 𝑦 = suc 𝑔)
126	125	reseq2d 5827	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑔 → (𝑢 ↾ suc 𝑦) = (𝑢 ↾ suc 𝑔))
127	125	reseq2d 5827	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑔 → (𝑣 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑔))
128	126, 127	eqeq12d 2774	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑔 → ((𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦) ↔ (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)))
129	128	imbi2d 344	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑔 → ((¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔))))
130	129	ralbidv 3126	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑔 → (∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔))))
131	124, 130	anbi12d 633	. . . . . . . . . 10 ⊢ (𝑦 = 𝑔 → ((𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) ↔ (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)))))
132	131	rexbidv 3221	. . . . . . . . 9 ⊢ (𝑦 = 𝑔 → (∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) ↔ ∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)))))
133	132	rexab 3611	. . . . . . . 8 ⊢ (∃𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}𝑧 = (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)) ↔ ∃𝑔(∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔))) ∧ 𝑧 = (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))
134		eqid 2758	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢‘𝑔) = (𝑢‘𝑔)
135		fvex 6675	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢‘𝑔) ∈ V
136		eqeq2 2770	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = (𝑢‘𝑔) → ((𝑢‘𝑔) = 𝑥 ↔ (𝑢‘𝑔) = (𝑢‘𝑔)))
137	136	3anbi3d 1439	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = (𝑢‘𝑔) → ((𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥) ↔ (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = (𝑢‘𝑔))))
138	135, 137	spcev 3527	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = (𝑢‘𝑔)) → ∃𝑥(𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))
139	134, 138	mp3an3 1447	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔))) → ∃𝑥(𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))
140	139	reximi 3171	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔))) → ∃𝑢 ∈ 𝐵 ∃𝑥(𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))
141		rexcom4 3177	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑢 ∈ 𝐵 ∃𝑥(𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥) ↔ ∃𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))
142	140, 141	sylib 221	. . . . . . . . . . . . . . 15 ⊢ (∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔))) → ∃𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))
143	142	adantl 485	. . . . . . . . . . . . . 14 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉)) ∧ ∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)))) → ∃𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))
144		noinfprefixmo 33493	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ⊆ No → ∃*𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))
145	144	ad2antrl 727	. . . . . . . . . . . . . . 15 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉)) → ∃*𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))
146	145	adantr 484	. . . . . . . . . . . . . 14 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉)) ∧ ∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)))) → ∃*𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))
147		df-eu 2588	. . . . . . . . . . . . . 14 ⊢ (∃!𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥) ↔ (∃𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥) ∧ ∃*𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))
148	143, 146, 147	sylanbrc 586	. . . . . . . . . . . . 13 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉)) ∧ ∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)))) → ∃!𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))
149		vex 3413	. . . . . . . . . . . . . 14 ⊢ 𝑧 ∈ V
150		eqeq2 2770	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑧 → ((𝑢‘𝑔) = 𝑥 ↔ (𝑢‘𝑔) = 𝑧))
151	150	3anbi3d 1439	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → ((𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥) ↔ (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑧)))
152	151	rexbidv 3221	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → (∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥) ↔ ∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑧)))
153	152	iota2 6328	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ V ∧ ∃!𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)) → (∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑧) ↔ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)) = 𝑧))
154	149, 153	mpan 689	. . . . . . . . . . . . 13 ⊢ (∃!𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥) → (∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑧) ↔ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)) = 𝑧))
155	148, 154	syl 17	. . . . . . . . . . . 12 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉)) ∧ ∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)))) → (∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑧) ↔ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)) = 𝑧))
156		eqcom 2765	. . . . . . . . . . . 12 ⊢ ((℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)) = 𝑧 ↔ 𝑧 = (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))
157	155, 156	bitrdi 290	. . . . . . . . . . 11 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉)) ∧ ∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)))) → (∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑧) ↔ 𝑧 = (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))
158		simprr3 1220	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ⊆ No ∧ (𝑢 ∈ 𝐵 ∧ (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑧))) → (𝑢‘𝑔) = 𝑧)
159		simpl 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵 ⊆ No ∧ (𝑢 ∈ 𝐵 ∧ (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑧))) → 𝐵 ⊆ No )
160		simprl 770	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵 ⊆ No ∧ (𝑢 ∈ 𝐵 ∧ (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑧))) → 𝑢 ∈ 𝐵)
161	159, 160	sseldd 3895	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐵 ⊆ No ∧ (𝑢 ∈ 𝐵 ∧ (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑧))) → 𝑢 ∈ No )
162		norn 33443	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ No → ran 𝑢 ⊆ {1_o, 2_o})
163	161, 162	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ⊆ No ∧ (𝑢 ∈ 𝐵 ∧ (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑧))) → ran 𝑢 ⊆ {1_o, 2_o})
164		nofun 33441	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 ∈ No → Fun 𝑢)
165	161, 164	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐵 ⊆ No ∧ (𝑢 ∈ 𝐵 ∧ (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑧))) → Fun 𝑢)
166		simprr1 1218	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐵 ⊆ No ∧ (𝑢 ∈ 𝐵 ∧ (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑧))) → 𝑔 ∈ dom 𝑢)
167		fvelrn 6840	. . . . . . . . . . . . . . . . 17 ⊢ ((Fun 𝑢 ∧ 𝑔 ∈ dom 𝑢) → (𝑢‘𝑔) ∈ ran 𝑢)
168	165, 166, 167	syl2anc 587	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ⊆ No ∧ (𝑢 ∈ 𝐵 ∧ (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑧))) → (𝑢‘𝑔) ∈ ran 𝑢)
169	163, 168	sseldd 3895	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ⊆ No ∧ (𝑢 ∈ 𝐵 ∧ (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑧))) → (𝑢‘𝑔) ∈ {1_o, 2_o})
170	158, 169	eqeltrrd 2853	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ⊆ No ∧ (𝑢 ∈ 𝐵 ∧ (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑧))) → 𝑧 ∈ {1_o, 2_o})
171	170	rexlimdvaa 3209	. . . . . . . . . . . . 13 ⊢ (𝐵 ⊆ No → (∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑧) → 𝑧 ∈ {1_o, 2_o}))
172	171	ad2antrl 727	. . . . . . . . . . . 12 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉)) → (∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑧) → 𝑧 ∈ {1_o, 2_o}))
173	172	adantr 484	. . . . . . . . . . 11 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉)) ∧ ∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)))) → (∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑧) → 𝑧 ∈ {1_o, 2_o}))
174	157, 173	sylbird 263	. . . . . . . . . 10 ⊢ (((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉)) ∧ ∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)))) → (𝑧 = (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)) → 𝑧 ∈ {1_o, 2_o}))
175	174	expimpd 457	. . . . . . . . 9 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉)) → ((∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔))) ∧ 𝑧 = (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))) → 𝑧 ∈ {1_o, 2_o}))
176	175	exlimdv 1934	. . . . . . . 8 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉)) → (∃𝑔(∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔))) ∧ 𝑧 = (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))) → 𝑧 ∈ {1_o, 2_o}))
177	133, 176	syl5bi 245	. . . . . . 7 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉)) → (∃𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}𝑧 = (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)) → 𝑧 ∈ {1_o, 2_o}))
178	177	abssdv 3975	. . . . . 6 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉)) → {𝑧 ∣ ∃𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}𝑧 = (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))} ⊆ {1_o, 2_o})
179	123, 178	eqsstrid 3942	. . . . 5 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉)) → ran (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))) ⊆ {1_o, 2_o})
180		elno2 33446	. . . . 5 ⊢ ((𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))) ∈ No ↔ (Fun (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))) ∧ dom (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))) ∈ On ∧ ran (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))) ⊆ {1_o, 2_o}))
181	21, 122, 179, 180	syl3anbrc 1340	. . . 4 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉)) → (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))) ∈ No )
182	19, 181	eqeltrd 2852	. . 3 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉)) → if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1_o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) ∈ No )
183	17, 182	pm2.61ian 811	. 2 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) → if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1_o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) ∈ No )
184	1, 183	eqeltrid 2856	1 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) → 𝑇 ∈ No )

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 ∀wal 1536 = wceq 1538 ∃wex 1781 ∈ wcel 2111 ∃*wmo 2555 ∃!weu 2587 {cab 2735 ∀wral 3070 ∃wrex 3071 ∃!wreu 3072 ∃*wrmo 3073 Vcvv 3409 ∪ cun 3858 ⊆ wss 3860 ifcif 4423 {csn 4525 {cpr 4527 ⟨cop 4531 ∪ cuni 4801 class class class wbr 5035 ↦ cmpt 5115 Tr wtr 5141 dom cdm 5527 ran crn 5528 ↾ cres 5529 “ cima 5530 Ord word 6172 Oncon0 6173 suc csuc 6175 ℩cio 6296 Fun wfun 6333 Fn wfn 6334 –onto→wfo 6337 ‘cfv 6339 ℩crio 7112 1_oc1o 8110 2_oc2o 8111 No csur 33432 <s cslt 33433 bday cbday 33434

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-rep 5159 ax-sep 5172 ax-nul 5179 ax-pr 5301 ax-un 7464

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-ral 3075 df-rex 3076 df-reu 3077 df-rmo 3078 df-rab 3079 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-pss 3879 df-nul 4228 df-if 4424 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4802 df-iun 4888 df-br 5036 df-opab 5098 df-mpt 5116 df-tr 5142 df-id 5433 df-eprel 5438 df-po 5446 df-so 5447 df-fr 5486 df-we 5488 df-xp 5533 df-rel 5534 df-cnv 5535 df-co 5536 df-dm 5537 df-rn 5538 df-res 5539 df-ima 5540 df-ord 6176 df-on 6177 df-suc 6179 df-iota 6298 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7113 df-1o 8117 df-2o 8118 df-no 33435 df-slt 33436 df-bday 33437

This theorem is referenced by: noinfbday 33512 noinfres 33514 noinfbnd1lem1 33515 noinfbnd1lem2 33516 noinfbnd1lem3 33517 noinfbnd1lem4 33518 noinfbnd1lem5 33519 noinfbnd1lem6 33520 noinfbnd2 33523 nosupinfsep 33524 noetainflem1 33529 noetainflem2 33530 noetainflem3 33531 noetainflem4 33532 noetalem1 33533

W3C validator