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Theorem noinfno 28102

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 4491	. . . . 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 486	. . . 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 783	. . . . . 6 ⊢ ((∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉)) → 𝐵 ⊆ No)
5		simpl 488	. . . . . . . 8 ⊢ ((∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉)) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥)
6		nominmo 28083	. . . . . . . . 9 ⊢ (𝐵 ⊆ No → ∃*𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥)
7	6	ad2antrl 741	. . . . . . . 8 ⊢ ((∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉)) → ∃*𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥)
8		reu5 3371	. . . . . . . 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 595	. . . . . . 7 ⊢ ((∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉)) → ∃!𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥)
10		riotacl 7414	. . . . . . 7 ⊢ (∃!𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 → (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∈ 𝐵)
11	9, 10	syl 18	. . . . . 6 ⊢ ((∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉)) → (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∈ 𝐵)
12	4, 11	sseldd 3914	. . . . 5 ⊢ ((∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉)) → (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∈ No)
13		1oex 8511	. . . . . . 7 ⊢ 1_o ∈ V
14	13	prid1 4723	. . . . . 6 ⊢ 1_o ∈ {1_o, 2_o}
15	14	noextend 28050	. . . . 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 18	. . . 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 2864	. . 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 4494	. . . . 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 486	. . . 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 6595	. . . . . 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 6530	. . . . . . 7 ⊢ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)) ∈ V
23		eqid 2763	. . . . . . 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 6702	. . . . . 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 488	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ⊆ No ∧ (𝑢 ∈ 𝐵 ∧ (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))))) → 𝐵 ⊆ No)
26		simprl 783	. . . . . . . . . . . . . 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 3914	. . . . . . . . . . . . 13 ⊢ ((𝐵 ⊆ No ∧ (𝑢 ∈ 𝐵 ∧ (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))))) → 𝑢 ∈ No)
28		nodmon 28034	. . . . . . . . . . . . 13 ⊢ (𝑢 ∈ No → dom 𝑢 ∈ On)
29	27, 28	syl 18	. . . . . . . . . . . 12 ⊢ ((𝐵 ⊆ No ∧ (𝑢 ∈ 𝐵 ∧ (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))))) → dom 𝑢 ∈ On)
30		simprrl 793	. . . . . . . . . . . 12 ⊢ ((𝐵 ⊆ No ∧ (𝑢 ∈ 𝐵 ∧ (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))))) → 𝑦 ∈ dom 𝑢)
31		onelon 6402	. . . . . . . . . . . 12 ⊢ ((dom 𝑢 ∈ On ∧ 𝑦 ∈ dom 𝑢) → 𝑦 ∈ On)
32	29, 30, 31	syl2anc 596	. . . . . . . . . . 11 ⊢ ((𝐵 ⊆ No ∧ (𝑢 ∈ 𝐵 ∧ (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))))) → 𝑦 ∈ On)
33	32	rexlimdvaa 3168	. . . . . . . . . 10 ⊢ (𝐵 ⊆ No → (∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) → 𝑦 ∈ On))
34	33	abssdv 3995	. . . . . . . . 9 ⊢ (𝐵 ⊆ No → {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ⊆ On)
35		simplr 781	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐵 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐵) → 𝑎 ∈ 𝑏)
36		ssel2 3908	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 ⊆ No ∧ 𝑢 ∈ 𝐵) → 𝑢 ∈ No)
37	36	adantlr 728	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐵 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐵) → 𝑢 ∈ No)
38	37, 28	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐵 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐵) → dom 𝑢 ∈ On)
39		ontr1 6425	. . . . . . . . . . . . . . . . . 18 ⊢ (dom 𝑢 ∈ On → ((𝑎 ∈ 𝑏 ∧ 𝑏 ∈ dom 𝑢) → 𝑎 ∈ dom 𝑢))
40	38, 39	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐵 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐵) → ((𝑎 ∈ 𝑏 ∧ 𝑏 ∈ dom 𝑢) → 𝑎 ∈ dom 𝑢))
41	35, 40	mpand 708	. . . . . . . . . . . . . . . 16 ⊢ (((𝐵 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐵) → (𝑏 ∈ dom 𝑢 → 𝑎 ∈ dom 𝑢))
42	41	adantrd 497	. . . . . . . . . . . . . . 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 5974	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏) → ((𝑢 ↾ suc 𝑏) ↾ suc 𝑎) = ((𝑣 ↾ suc 𝑏) ↾ suc 𝑎))
44		onelon 6402	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((dom 𝑢 ∈ On ∧ 𝑏 ∈ dom 𝑢) → 𝑏 ∈ On)
45	38, 44	sylan 592	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐵 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐵) ∧ 𝑏 ∈ dom 𝑢) → 𝑏 ∈ On)
46		onsucb 7849	. . . . . . . . . . . . . . . . . . . . . . 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 788	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐵 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐵) ∧ 𝑏 ∈ dom 𝑢) → 𝑎 ∈ 𝑏)
49		eloni 6387	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑏 ∈ On → Ord 𝑏)
50	45, 49	syl 18	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐵 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐵) ∧ 𝑏 ∈ dom 𝑢) → Ord 𝑏)
51		ordsucelsuc 7854	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (Ord 𝑏 → (𝑎 ∈ 𝑏 ↔ suc 𝑎 ∈ suc 𝑏))
52	50, 51	syl 18	. . . . . . . . . . . . . . . . . . . . . . 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 6420	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (suc 𝑏 ∈ On → (suc 𝑎 ∈ suc 𝑏 → suc 𝑎 ⊆ suc 𝑏))
55	47, 53, 54	sylc 66	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐵 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐵) ∧ 𝑏 ∈ dom 𝑢) → suc 𝑎 ⊆ suc 𝑏)
56	55	resabs1d 6010	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐵 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐵) ∧ 𝑏 ∈ dom 𝑢) → ((𝑢 ↾ suc 𝑏) ↾ suc 𝑎) = (𝑢 ↾ suc 𝑎))
57	55	resabs1d 6010	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐵 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐵) ∧ 𝑏 ∈ dom 𝑢) → ((𝑣 ↾ suc 𝑏) ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎))
58	56, 57	eqeq12d 2779	. . . . . . . . . . . . . . . . . . 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	imbitrid 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 58	. . . . . . . . . . . . . . . . 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 3180	. . . . . . . . . . . . . . . 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 459	. . . . . . . . . . . . . . 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 522	. . . . . . . . . . . . . 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 3179	. . . . . . . . . . . . 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 459	. . . . . . . . . . . 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 3458	. . . . . . . . . . . . . 14 ⊢ 𝑏 ∈ V
67		eleq1w 2847	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑏 → (𝑦 ∈ dom 𝑢 ↔ 𝑏 ∈ dom 𝑢))
68		suceq 6446	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑏 → suc 𝑦 = suc 𝑏)
69	68	reseq2d 5980	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑏 → (𝑢 ↾ suc 𝑦) = (𝑢 ↾ suc 𝑏))
70	68	reseq2d 5980	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑏 → (𝑣 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑏))
71	69, 70	eqeq12d 2779	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑏 → ((𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦) ↔ (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏)))
72	71	imbi2d 343	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑏 → ((¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏))))
73	72	ralbidv 3189	. . . . . . . . . . . . . . . 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 644	. . . . . . . . . . . . . . 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 3190	. . . . . . . . . . . . . 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 3636	. . . . . . . . . . . . 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 635	. . . . . . . . . . . 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 3458	. . . . . . . . . . . . 13 ⊢ 𝑎 ∈ V
79		eleq1w 2847	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑎 → (𝑦 ∈ dom 𝑢 ↔ 𝑎 ∈ dom 𝑢))
80		suceq 6446	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑎 → suc 𝑦 = suc 𝑎)
81	80	reseq2d 5980	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑎 → (𝑢 ↾ suc 𝑦) = (𝑢 ↾ suc 𝑎))
82	80	reseq2d 5980	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑎 → (𝑣 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑎))
83	81, 82	eqeq12d 2779	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑎 → ((𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦) ↔ (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎)))
84	83	imbi2d 343	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑎 → ((¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎))))
85	84	ralbidv 3189	. . . . . . . . . . . . . . 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 644	. . . . . . . . . . . . . 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 3190	. . . . . . . . . . . . 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 3636	. . . . . . . . . . . 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 1963	. . . . . . . . . 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 5217	. . . . . . . . . 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 7819	. . . . . . . . 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 595	. . . . . . . 8 ⊢ (𝐵 ⊆ No → Ord {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))})
95	94	ad2antrl 741	. . . . . . 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 28061	. . . . . . . . . . . . . 14 ⊢ bday:No⟶ontoOn
97		fofun 6816	. . . . . . . . . . . . . 14 ⊢ (bday:No⟶ontoOn → Fun bday)
98	96, 97	ax-mp 5	. . . . . . . . . . . . 13 ⊢ Fun bday
99		funimaexg 6645	. . . . . . . . . . . . 13 ⊢ ((Fun bday ∧ 𝐵 ∈ 𝑉) → (bday “ 𝐵) ∈ V)
100	98, 99	mpan 703	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ 𝑉 → (bday “ 𝐵) ∈ V)
101	100	uniexd 7780	. . . . . . . . . . 11 ⊢ (𝐵 ∈ 𝑉 → ∪ (bday “ 𝐵) ∈ V)
102	101	adantl 487	. . . . . . . . . 10 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) → ∪ (bday “ 𝐵) ∈ V)
103	102	adantl 487	. . . . . . . . 9 ⊢ ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉)) → ∪ (bday “ 𝐵) ∈ V)
104		bdayval 28032	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ No → (bday‘𝑢) = dom 𝑢)
105	27, 104	syl 18	. . . . . . . . . . . . . . 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 6817	. . . . . . . . . . . . . . . . . 18 ⊢ (bday:No⟶ontoOn → bday Fn No)
107	96, 106	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ bday Fn No
108		fnfvima 7259	. . . . . . . . . . . . . . . . 17 ⊢ ((bday Fn No ∧ 𝐵 ⊆ No ∧ 𝑢 ∈ 𝐵) → (bday‘𝑢) ∈ (bday “ 𝐵))
109	107, 108	mp3an1 1478	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ⊆ No ∧ 𝑢 ∈ 𝐵) → (bday‘𝑢) ∈ (bday “ 𝐵))
110	109	adantrr 730	. . . . . . . . . . . . . . 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 2865	. . . . . . . . . . . . . 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 4902	. . . . . . . . . . . . . 14 ⊢ (dom 𝑢 ∈ (bday “ 𝐵) → dom 𝑢 ⊆ ∪ (bday “ 𝐵))
113	111, 112	syl 18	. . . . . . . . . . . . 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 3914	. . . . . . . . . . . 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 3168	. . . . . . . . . . 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 3995	. . . . . . . . . 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 741	. . . . . . . . 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 5290	. . . . . . . 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 6385	. . . . . . . 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 18	. . . . . . 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 2868	. . . . 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 5945	. . . . . 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 2847	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑔 → (𝑦 ∈ dom 𝑢 ↔ 𝑔 ∈ dom 𝑢))
125		suceq 6446	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑔 → suc 𝑦 = suc 𝑔)
126	125	reseq2d 5980	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑔 → (𝑢 ↾ suc 𝑦) = (𝑢 ↾ suc 𝑔))
127	125	reseq2d 5980	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑔 → (𝑣 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑔))
128	126, 127	eqeq12d 2779	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑔 → ((𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦) ↔ (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)))
129	128	imbi2d 343	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑔 → ((¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔))))
130	129	ralbidv 3189	. . . . . . . . . . 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 644	. . . . . . . . . 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 3190	. . . . . . . . 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 3656	. . . . . . . 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 2763	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢‘𝑔) = (𝑢‘𝑔)
135		fvex 6918	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢‘𝑔) ∈ V
136		eqeq2 2775	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = (𝑢‘𝑔) → ((𝑢‘𝑔) = 𝑥 ↔ (𝑢‘𝑔) = (𝑢‘𝑔)))
137	136	3anbi3d 1471	. . . . . . . . . . . . . . . . . . 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 3564	. . . . . . . . . . . . . . . . . 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 1480	. . . . . . . . . . . . . . . . 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 3104	. . . . . . . . . . . . . . . 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 3293	. . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . 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 28085	. . . . . . . . . . . . . . . 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 741	. . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . 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 2597	. . . . . . . . . . . . . 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 595	. . . . . . . . . . . . 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 3458	. . . . . . . . . . . . . 14 ⊢ 𝑧 ∈ V
150		eqeq2 2775	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑧 → ((𝑢‘𝑔) = 𝑥 ↔ (𝑢‘𝑔) = 𝑧))
151	150	3anbi3d 1471	. . . . . . . . . . . . . . . 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 3190	. . . . . . . . . . . . . . 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 6543	. . . . . . . . . . . . . 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 703	. . . . . . . . . . . . 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 18	. . . . . . . . . . . 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 2770	. . . . . . . . . . . 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 1242	. . . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . . . . 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 783	. . . . . . . . . . . . . . . . . 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 3914	. . . . . . . . . . . . . . . . 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 28035	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ No → ran 𝑢 ⊆ {1_o, 2_o})
163	161, 162	syl 18	. . . . . . . . . . . . . . . 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 28033	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 ∈ No → Fun 𝑢)
165	161, 164	syl 18	. . . . . . . . . . . . . . . . 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 1240	. . . . . . . . . . . . . . . . 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 7097	. . . . . . . . . . . . . . . . 17 ⊢ ((Fun 𝑢 ∧ 𝑔 ∈ dom 𝑢) → (𝑢‘𝑔) ∈ ran 𝑢)
168	165, 166, 167	syl2anc 596	. . . . . . . . . . . . . . . 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 3914	. . . . . . . . . . . . . . 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 2865	. . . . . . . . . . . . . 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 3168	. . . . . . . . . . . . 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 741	. . . . . . . . . . . 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 486	. . . . . . . . . . 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 459	. . . . . . . . 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 1968	. . . . . . . 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	biimtrid 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 3995	. . . . . 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 3951	. . . . 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 28038	. . . . 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 1362	. . . 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 2864	. . 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 824	. 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 2868	1 ⊢ ((𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) → 𝑇 ∈ No)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 ∀wal 1568 = wceq 1570 ∃wex 1812 ∈ wcel 2147 ∃*wmo 2565 ∃!weu 2596 {cab 2741 ∀wral 3080 ∃wrex 3090 ∃!wreu 3367 ∃*wrmo 3368 Vcvv 3454 ⊆ wss 3900 ∪ cun 4152 ifcif 4485 {csn 4588 {cpr 4590 ⟨cop 4838 ∪ cuni 4870 class class class wbr 5106 ↦ cmpt 5189 Tr wtr 5215 dom cdm 5655 ran crn 5656 ↾ cres 5657 “ cima 5658 Ord word 6376 Oncon0 6377 suc csuc 6379 ℩cio 6507 Fun wfun 6549 Fn wfn 6550 ⟶ontowfo 6553 ‘cfv 6555 ℩crio 7396 1_oc1o 8494 2_oc2o 8495 Nocsur 28024 <s clts 28025 bdaycbday 28026

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7772

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-ord 6380 df-on 6381 df-suc 6383 df-iota 6509 df-fun 6557 df-fn 6558 df-f 6559 df-fo 6561 df-fv 6563 df-riota 7397 df-1o 8501 df-2o 8502 df-no 28027 df-lts 28028 df-bday 28029

This theorem is used by: noinfbday 28104 noinfres 28106 noinfbnd1lem1 28107 noinfbnd1lem2 28108 noinfbnd1lem3 28109 noinfbnd1lem4 28110 noinfbnd1lem5 28111 noinfbnd1lem6 28112 noinfbnd2 28115 nosupinfsep 28116 noetainflem1 28121 noetainflem2 28122 noetainflem3 28123 noetainflem4 28124 noetalem1 28125

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