nosupno - Mathbox for Scott Fenton

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Theorem nosupno 33906

Description: The next several theorems deal with a surreal "supremum". This surreal will ultimately be shown to bound 𝐴 below and bound the restriction of any surreal above. We begin by showing that the given expression actually defines a surreal number. (Contributed by Scott Fenton, 5-Dec-2021.)

**Hypothesis**
Ref	Expression
nosupno.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
nosupno	⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ 𝑉) → 𝑆 ∈ No )

Distinct variable group: 𝑥,𝐴,𝑦,𝑔,𝑣,𝑢 Allowed substitution hints: 𝑆(𝑥,𝑦,𝑣,𝑢,𝑔) 𝑉(𝑥,𝑦,𝑣,𝑢,𝑔)

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

Step	Hyp	Ref	Expression
1		elex 3450	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		nosupno.1	. . 3 ⊢ 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))))
3		iftrue 4465	. . . . . 6 ⊢ (∃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⟩}))
4	3	adantr 481	. . . . 5 ⊢ ((∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V)) → 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⟩}))
5		simprl 768	. . . . . . 7 ⊢ ((∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V)) → 𝐴 ⊆ No )
6		simpl 483	. . . . . . . . 9 ⊢ ((∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V)) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
7		nomaxmo 33901	. . . . . . . . . 10 ⊢ (𝐴 ⊆ No → ∃*𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
8	7	ad2antrl 725	. . . . . . . . 9 ⊢ ((∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V)) → ∃*𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
9		reu5 3361	. . . . . . . . 9 ⊢ (∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ↔ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ∃*𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦))
10	6, 8, 9	sylanbrc 583	. . . . . . . 8 ⊢ ((∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V)) → ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
11		riotacl 7250	. . . . . . . 8 ⊢ (∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 → (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∈ 𝐴)
12	10, 11	syl 17	. . . . . . 7 ⊢ ((∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V)) → (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∈ 𝐴)
13	5, 12	sseldd 3922	. . . . . 6 ⊢ ((∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V)) → (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∈ No )
14		2on 8311	. . . . . . . . 9 ⊢ 2_o ∈ On
15	14	elexi 3451	. . . . . . . 8 ⊢ 2_o ∈ V
16	15	prid2 4699	. . . . . . 7 ⊢ 2_o ∈ {1_o, 2_o}
17	16	noextend 33869	. . . . . 6 ⊢ ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∈ No → ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2_o⟩}) ∈ No )
18	13, 17	syl 17	. . . . 5 ⊢ ((∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V)) → ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2_o⟩}) ∈ No )
19	4, 18	eqeltrd 2839	. . . 4 ⊢ ((∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V)) → 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)))) ∈ No )
20		iffalse 4468	. . . . . 6 ⊢ (¬ ∃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)))) = (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))))
21	20	adantr 481	. . . . 5 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V)) → 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)))) = (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))))
22		funmpt 6472	. . . . . . 7 ⊢ Fun (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)))
23	22	a1i 11	. . . . . 6 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V)) → Fun (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))))
24		iotaex 6413	. . . . . . . . 9 ⊢ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)) ∈ V
25		eqid 2738	. . . . . . . . 9 ⊢ (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))) = (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)))
26	24, 25	dmmpti 6577	. . . . . . . 8 ⊢ dom (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))) = {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}
27		ssel2 3916	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ⊆ No ∧ 𝑢 ∈ 𝐴) → 𝑢 ∈ No )
28		nodmon 33853	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ No → dom 𝑢 ∈ On)
29	27, 28	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ⊆ No ∧ 𝑢 ∈ 𝐴) → dom 𝑢 ∈ On)
30		onss 7634	. . . . . . . . . . . . . . . 16 ⊢ (dom 𝑢 ∈ On → dom 𝑢 ⊆ On)
31	29, 30	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ No ∧ 𝑢 ∈ 𝐴) → dom 𝑢 ⊆ On)
32	31	sseld 3920	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ No ∧ 𝑢 ∈ 𝐴) → (𝑦 ∈ dom 𝑢 → 𝑦 ∈ On))
33	32	adantrd 492	. . . . . . . . . . . . 13 ⊢ ((𝐴 ⊆ No ∧ 𝑢 ∈ 𝐴) → ((𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) → 𝑦 ∈ On))
34	33	rexlimdva 3213	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ No → (∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) → 𝑦 ∈ On))
35	34	abssdv 4002	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ No → {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ⊆ On)
36		simplr 766	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐴) → 𝑎 ∈ 𝑏)
37	29	adantlr 712	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐴) → dom 𝑢 ∈ On)
38		ontr1 6312	. . . . . . . . . . . . . . . . . . . 20 ⊢ (dom 𝑢 ∈ On → ((𝑎 ∈ 𝑏 ∧ 𝑏 ∈ dom 𝑢) → 𝑎 ∈ dom 𝑢))
39	37, 38	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐴) → ((𝑎 ∈ 𝑏 ∧ 𝑏 ∈ dom 𝑢) → 𝑎 ∈ dom 𝑢))
40	36, 39	mpand 692	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐴) → (𝑏 ∈ dom 𝑢 → 𝑎 ∈ dom 𝑢))
41	40	adantrd 492	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐴) → ((𝑏 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏))) → 𝑎 ∈ dom 𝑢))
42		reseq1 5885	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏) → ((𝑢 ↾ suc 𝑏) ↾ suc 𝑎) = ((𝑣 ↾ suc 𝑏) ↾ suc 𝑎))
43		onelon 6291	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((dom 𝑢 ∈ On ∧ 𝑏 ∈ dom 𝑢) → 𝑏 ∈ On)
44	37, 43	sylan 580	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐴 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐴) ∧ 𝑏 ∈ dom 𝑢) → 𝑏 ∈ On)
45		suceloni 7659	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑏 ∈ On → suc 𝑏 ∈ On)
46	44, 45	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐴 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐴) ∧ 𝑏 ∈ dom 𝑢) → suc 𝑏 ∈ On)
47		simpllr 773	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐴 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐴) ∧ 𝑏 ∈ dom 𝑢) → 𝑎 ∈ 𝑏)
48		eloni 6276	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑏 ∈ On → Ord 𝑏)
49	44, 48	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐴 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐴) ∧ 𝑏 ∈ dom 𝑢) → Ord 𝑏)
50		ordsucelsuc 7669	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (Ord 𝑏 → (𝑎 ∈ 𝑏 ↔ suc 𝑎 ∈ suc 𝑏))
51	49, 50	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐴 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐴) ∧ 𝑏 ∈ dom 𝑢) → (𝑎 ∈ 𝑏 ↔ suc 𝑎 ∈ suc 𝑏))
52	47, 51	mpbid 231	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐴 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐴) ∧ 𝑏 ∈ dom 𝑢) → suc 𝑎 ∈ suc 𝑏)
53		onelss 6308	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (suc 𝑏 ∈ On → (suc 𝑎 ∈ suc 𝑏 → suc 𝑎 ⊆ suc 𝑏))
54	46, 52, 53	sylc 65	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐴 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐴) ∧ 𝑏 ∈ dom 𝑢) → suc 𝑎 ⊆ suc 𝑏)
55	54	resabs1d 5922	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐴 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐴) ∧ 𝑏 ∈ dom 𝑢) → ((𝑢 ↾ suc 𝑏) ↾ suc 𝑎) = (𝑢 ↾ suc 𝑎))
56	54	resabs1d 5922	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐴 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐴) ∧ 𝑏 ∈ dom 𝑢) → ((𝑣 ↾ suc 𝑏) ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎))
57	55, 56	eqeq12d 2754	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐴 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐴) ∧ 𝑏 ∈ dom 𝑢) → (((𝑢 ↾ suc 𝑏) ↾ suc 𝑎) = ((𝑣 ↾ suc 𝑏) ↾ suc 𝑎) ↔ (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎)))
58	42, 57	syl5ib 243	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐴 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐴) ∧ 𝑏 ∈ dom 𝑢) → ((𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏) → (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎)))
59	58	imim2d 57	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐴 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐴) ∧ 𝑏 ∈ dom 𝑢) → ((¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏)) → (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎))))
60	59	ralimdv 3109	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐴) ∧ 𝑏 ∈ dom 𝑢) → (∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏)) → ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎))))
61	60	expimpd 454	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐴) → ((𝑏 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏))) → ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎))))
62	41, 61	jcad 513	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐴) → ((𝑏 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏))) → (𝑎 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎)))))
63	62	reximdva 3203	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ No ∧ 𝑎 ∈ 𝑏) → (∃𝑢 ∈ 𝐴 (𝑏 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏))) → ∃𝑢 ∈ 𝐴 (𝑎 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎)))))
64	63	expimpd 454	. . . . . . . . . . . . . 14 ⊢ (𝐴 ⊆ No → ((𝑎 ∈ 𝑏 ∧ ∃𝑢 ∈ 𝐴 (𝑏 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏)))) → ∃𝑢 ∈ 𝐴 (𝑎 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎)))))
65		vex 3436	. . . . . . . . . . . . . . . 16 ⊢ 𝑏 ∈ V
66		eleq1w 2821	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑏 → (𝑦 ∈ dom 𝑢 ↔ 𝑏 ∈ dom 𝑢))
67		suceq 6331	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = 𝑏 → suc 𝑦 = suc 𝑏)
68	67	reseq2d 5891	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑏 → (𝑢 ↾ suc 𝑦) = (𝑢 ↾ suc 𝑏))
69	67	reseq2d 5891	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑏 → (𝑣 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑏))
70	68, 69	eqeq12d 2754	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑏 → ((𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦) ↔ (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏)))
71	70	imbi2d 341	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑏 → ((¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏))))
72	71	ralbidv 3112	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑏 → (∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏))))
73	66, 72	anbi12d 631	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑏 → ((𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) ↔ (𝑏 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏)))))
74	73	rexbidv 3226	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑏 → (∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) ↔ ∃𝑢 ∈ 𝐴 (𝑏 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏)))))
75	65, 74	elab 3609	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↔ ∃𝑢 ∈ 𝐴 (𝑏 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏))))
76	75	anbi2i 623	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ 𝑏 ∧ 𝑏 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}) ↔ (𝑎 ∈ 𝑏 ∧ ∃𝑢 ∈ 𝐴 (𝑏 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏)))))
77		vex 3436	. . . . . . . . . . . . . . 15 ⊢ 𝑎 ∈ V
78		eleq1w 2821	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑎 → (𝑦 ∈ dom 𝑢 ↔ 𝑎 ∈ dom 𝑢))
79		suceq 6331	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑎 → suc 𝑦 = suc 𝑎)
80	79	reseq2d 5891	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑎 → (𝑢 ↾ suc 𝑦) = (𝑢 ↾ suc 𝑎))
81	79	reseq2d 5891	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑎 → (𝑣 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑎))
82	80, 81	eqeq12d 2754	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑎 → ((𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦) ↔ (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎)))
83	82	imbi2d 341	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑎 → ((¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎))))
84	83	ralbidv 3112	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑎 → (∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎))))
85	78, 84	anbi12d 631	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑎 → ((𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) ↔ (𝑎 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎)))))
86	85	rexbidv 3226	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑎 → (∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) ↔ ∃𝑢 ∈ 𝐴 (𝑎 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎)))))
87	77, 86	elab 3609	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↔ ∃𝑢 ∈ 𝐴 (𝑎 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎))))
88	64, 76, 87	3imtr4g 296	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ No → ((𝑎 ∈ 𝑏 ∧ 𝑏 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}) → 𝑎 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}))
89	88	alrimivv 1931	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ No → ∀𝑎∀𝑏((𝑎 ∈ 𝑏 ∧ 𝑏 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}) → 𝑎 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}))
90		dftr2 5193	. . . . . . . . . . . 12 ⊢ (Tr {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↔ ∀𝑎∀𝑏((𝑎 ∈ 𝑏 ∧ 𝑏 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}) → 𝑎 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}))
91	89, 90	sylibr 233	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ No → Tr {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))})
92		dford5 33671	. . . . . . . . . . 11 ⊢ (Ord {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↔ ({𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ⊆ On ∧ Tr {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}))
93	35, 91, 92	sylanbrc 583	. . . . . . . . . 10 ⊢ (𝐴 ⊆ No → Ord {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))})
94	93	adantr 481	. . . . . . . . 9 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V) → Ord {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))})
95		bdayfo 33880	. . . . . . . . . . . . . . 15 ⊢ bday : No –onto→On
96		fofun 6689	. . . . . . . . . . . . . . 15 ⊢ ( bday : No –onto→On → Fun bday )
97	95, 96	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ Fun bday
98		funimaexg 6520	. . . . . . . . . . . . . 14 ⊢ ((Fun bday ∧ 𝐴 ∈ V) → ( bday “ 𝐴) ∈ V)
99	97, 98	mpan 687	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ V → ( bday “ 𝐴) ∈ V)
100	99	uniexd 7595	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ V → ∪ ( bday “ 𝐴) ∈ V)
101	100	adantl 482	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V) → ∪ ( bday “ 𝐴) ∈ V)
102		simpl 483	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) → 𝑦 ∈ dom 𝑢)
103	102	reximi 3178	. . . . . . . . . . . . 13 ⊢ (∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) → ∃𝑢 ∈ 𝐴 𝑦 ∈ dom 𝑢)
104	103	ss2abi 4000	. . . . . . . . . . . 12 ⊢ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ⊆ {𝑦 ∣ ∃𝑢 ∈ 𝐴 𝑦 ∈ dom 𝑢}
105		bdayval 33851	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 ∈ No → ( bday ‘𝑢) = dom 𝑢)
106	27, 105	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ⊆ No ∧ 𝑢 ∈ 𝐴) → ( bday ‘𝑢) = dom 𝑢)
107		fofn 6690	. . . . . . . . . . . . . . . . . . . 20 ⊢ ( bday : No –onto→On → bday Fn No )
108	95, 107	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ bday Fn No
109		fnfvima 7109	. . . . . . . . . . . . . . . . . . 19 ⊢ (( bday Fn No ∧ 𝐴 ⊆ No ∧ 𝑢 ∈ 𝐴) → ( bday ‘𝑢) ∈ ( bday “ 𝐴))
110	108, 109	mp3an1 1447	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ⊆ No ∧ 𝑢 ∈ 𝐴) → ( bday ‘𝑢) ∈ ( bday “ 𝐴))
111	106, 110	eqeltrrd 2840	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ⊆ No ∧ 𝑢 ∈ 𝐴) → dom 𝑢 ∈ ( bday “ 𝐴))
112		elssuni 4871	. . . . . . . . . . . . . . . . 17 ⊢ (dom 𝑢 ∈ ( bday “ 𝐴) → dom 𝑢 ⊆ ∪ ( bday “ 𝐴))
113	111, 112	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ⊆ No ∧ 𝑢 ∈ 𝐴) → dom 𝑢 ⊆ ∪ ( bday “ 𝐴))
114	113	sseld 3920	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ No ∧ 𝑢 ∈ 𝐴) → (𝑦 ∈ dom 𝑢 → 𝑦 ∈ ∪ ( bday “ 𝐴)))
115	114	rexlimdva 3213	. . . . . . . . . . . . . 14 ⊢ (𝐴 ⊆ No → (∃𝑢 ∈ 𝐴 𝑦 ∈ dom 𝑢 → 𝑦 ∈ ∪ ( bday “ 𝐴)))
116	115	abssdv 4002	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ No → {𝑦 ∣ ∃𝑢 ∈ 𝐴 𝑦 ∈ dom 𝑢} ⊆ ∪ ( bday “ 𝐴))
117	116	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V) → {𝑦 ∣ ∃𝑢 ∈ 𝐴 𝑦 ∈ dom 𝑢} ⊆ ∪ ( bday “ 𝐴))
118	104, 117	sstrid 3932	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V) → {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ⊆ ∪ ( bday “ 𝐴))
119	101, 118	ssexd 5248	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V) → {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ∈ V)
120		elong 6274	. . . . . . . . . 10 ⊢ ({𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ∈ V → ({𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ∈ On ↔ Ord {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}))
121	119, 120	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V) → ({𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ∈ On ↔ Ord {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}))
122	94, 121	mpbird 256	. . . . . . . 8 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V) → {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ∈ On)
123	26, 122	eqeltrid 2843	. . . . . . 7 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V) → dom (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))) ∈ On)
124	123	adantl 482	. . . . . 6 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V)) → dom (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))) ∈ On)
125	25	rnmpt 5864	. . . . . . . 8 ⊢ ran (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))) = {𝑧 ∣ ∃𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}𝑧 = (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))}
126		vex 3436	. . . . . . . . . . . 12 ⊢ 𝑔 ∈ V
127		eleq1w 2821	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑔 → (𝑦 ∈ dom 𝑢 ↔ 𝑔 ∈ dom 𝑢))
128		suceq 6331	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑔 → suc 𝑦 = suc 𝑔)
129	128	reseq2d 5891	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑔 → (𝑢 ↾ suc 𝑦) = (𝑢 ↾ suc 𝑔))
130	128	reseq2d 5891	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑔 → (𝑣 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑔))
131	129, 130	eqeq12d 2754	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑔 → ((𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦) ↔ (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)))
132	131	imbi2d 341	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑔 → ((¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔))))
133	132	ralbidv 3112	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑔 → (∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔))))
134	127, 133	anbi12d 631	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑔 → ((𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) ↔ (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)))))
135	134	rexbidv 3226	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑔 → (∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) ↔ ∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)))))
136	126, 135	elab 3609	. . . . . . . . . . 11 ⊢ (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↔ ∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔))))
137		eqid 2738	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢‘𝑔) = (𝑢‘𝑔)
138		fvex 6787	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢‘𝑔) ∈ V
139		eqeq2 2750	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = (𝑢‘𝑔) → ((𝑢‘𝑔) = 𝑥 ↔ (𝑢‘𝑔) = (𝑢‘𝑔)))
140	139	3anbi3d 1441	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = (𝑢‘𝑔) → ((𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥) ↔ (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = (𝑢‘𝑔))))
141	138, 140	spcev 3545	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = (𝑢‘𝑔)) → ∃𝑥(𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))
142	137, 141	mp3an3 1449	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔))) → ∃𝑥(𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))
143	142	reximi 3178	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔))) → ∃𝑢 ∈ 𝐴 ∃𝑥(𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))
144		rexcom4 3233	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑢 ∈ 𝐴 ∃𝑥(𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥) ↔ ∃𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))
145	143, 144	sylib 217	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔))) → ∃𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))
146	145	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ No ∧ ∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)))) → ∃𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))
147		nosupprefixmo 33903	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ⊆ No → ∃*𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))
148	147	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ No ∧ ∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)))) → ∃*𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))
149		df-eu 2569	. . . . . . . . . . . . . . 15 ⊢ (∃!𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥) ↔ (∃𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥) ∧ ∃*𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))
150	146, 148, 149	sylanbrc 583	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ No ∧ ∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)))) → ∃!𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))
151		vex 3436	. . . . . . . . . . . . . . 15 ⊢ 𝑧 ∈ V
152		eqeq2 2750	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑧 → ((𝑢‘𝑔) = 𝑥 ↔ (𝑢‘𝑔) = 𝑧))
153	152	3anbi3d 1441	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑧 → ((𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥) ↔ (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑧)))
154	153	rexbidv 3226	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → (∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥) ↔ ∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑧)))
155	154	iota2 6422	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ V ∧ ∃!𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)) → (∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑧) ↔ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)) = 𝑧))
156	151, 155	mpan 687	. . . . . . . . . . . . . 14 ⊢ (∃!𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥) → (∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑧) ↔ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)) = 𝑧))
157	150, 156	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝐴 ⊆ No ∧ ∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)))) → (∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑧) ↔ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)) = 𝑧))
158		eqcom 2745	. . . . . . . . . . . . 13 ⊢ ((℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)) = 𝑧 ↔ 𝑧 = (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))
159	157, 158	bitrdi 287	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ No ∧ ∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)))) → (∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑧) ↔ 𝑧 = (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))
160		simprr3 1222	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ No ∧ (𝑢 ∈ 𝐴 ∧ (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑧))) → (𝑢‘𝑔) = 𝑧)
161	27	adantrr 714	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ⊆ No ∧ (𝑢 ∈ 𝐴 ∧ (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑧))) → 𝑢 ∈ No )
162		norn 33854	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ No → ran 𝑢 ⊆ {1_o, 2_o})
163	161, 162	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ⊆ No ∧ (𝑢 ∈ 𝐴 ∧ (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑧))) → ran 𝑢 ⊆ {1_o, 2_o})
164		nofun 33852	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 ∈ No → Fun 𝑢)
165	161, 164	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ⊆ No ∧ (𝑢 ∈ 𝐴 ∧ (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑧))) → Fun 𝑢)
166		simprr1 1220	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ⊆ No ∧ (𝑢 ∈ 𝐴 ∧ (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑧))) → 𝑔 ∈ dom 𝑢)
167		fvelrn 6954	. . . . . . . . . . . . . . . . 17 ⊢ ((Fun 𝑢 ∧ 𝑔 ∈ dom 𝑢) → (𝑢‘𝑔) ∈ ran 𝑢)
168	165, 166, 167	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ⊆ No ∧ (𝑢 ∈ 𝐴 ∧ (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑧))) → (𝑢‘𝑔) ∈ ran 𝑢)
169	163, 168	sseldd 3922	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ No ∧ (𝑢 ∈ 𝐴 ∧ (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑧))) → (𝑢‘𝑔) ∈ {1_o, 2_o})
170	160, 169	eqeltrrd 2840	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ No ∧ (𝑢 ∈ 𝐴 ∧ (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑧))) → 𝑧 ∈ {1_o, 2_o})
171	170	rexlimdvaa 3214	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ No → (∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑧) → 𝑧 ∈ {1_o, 2_o}))
172	171	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ No ∧ ∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)))) → (∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑧) → 𝑧 ∈ {1_o, 2_o}))
173	159, 172	sylbird 259	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ No ∧ ∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)))) → (𝑧 = (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)) → 𝑧 ∈ {1_o, 2_o}))
174	136, 173	sylan2b 594	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ No ∧ 𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}) → (𝑧 = (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)) → 𝑧 ∈ {1_o, 2_o}))
175	174	rexlimdva 3213	. . . . . . . . 9 ⊢ (𝐴 ⊆ No → (∃𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}𝑧 = (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)) → 𝑧 ∈ {1_o, 2_o}))
176	175	abssdv 4002	. . . . . . . 8 ⊢ (𝐴 ⊆ No → {𝑧 ∣ ∃𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))}𝑧 = (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))} ⊆ {1_o, 2_o})
177	125, 176	eqsstrid 3969	. . . . . . 7 ⊢ (𝐴 ⊆ No → ran (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))) ⊆ {1_o, 2_o})
178	177	ad2antrl 725	. . . . . 6 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V)) → ran (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))) ⊆ {1_o, 2_o})
179		elno2 33857	. . . . . 6 ⊢ ((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))) ∈ No ↔ (Fun (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))) ∧ dom (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))) ∈ On ∧ ran (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))) ⊆ {1_o, 2_o}))
180	23, 124, 178, 179	syl3anbrc 1342	. . . . 5 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V)) → (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))) ∈ No )
181	21, 180	eqeltrd 2839	. . . 4 ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V)) → 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)))) ∈ No )
182	19, 181	pm2.61ian 809	. . 3 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V) → 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)))) ∈ No )
183	2, 182	eqeltrid 2843	. 2 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V) → 𝑆 ∈ No )
184	1, 183	sylan2 593	1 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ 𝑉) → 𝑆 ∈ No )

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 ∀wal 1537 = wceq 1539 ∃wex 1782 ∈ wcel 2106 ∃*wmo 2538 ∃!weu 2568 {cab 2715 ∀wral 3064 ∃wrex 3065 ∃!wreu 3066 ∃*wrmo 3067 Vcvv 3432 ∪ cun 3885 ⊆ wss 3887 ifcif 4459 {csn 4561 {cpr 4563 ⟨cop 4567 ∪ cuni 4839 class class class wbr 5074 ↦ cmpt 5157 Tr wtr 5191 dom cdm 5589 ran crn 5590 ↾ cres 5591 “ cima 5592 Ord word 6265 Oncon0 6266 suc csuc 6268 ℩cio 6389 Fun wfun 6427 Fn wfn 6428 –onto→wfo 6431 ‘cfv 6433 ℩crio 7231 1_oc1o 8290 2_oc2o 8291 No csur 33843 <s cslt 33844 bday cbday 33845

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-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: nosupbday 33908 nosupres 33910 nosupbnd1lem1 33911 nosupbnd1lem2 33912 nosupbnd1lem3 33913 nosupbnd1lem4 33914 nosupbnd1lem5 33915 nosupbnd1lem6 33916 nosupbnd2 33919 nosupinfsep 33935 noetasuplem1 33936 noetasuplem2 33937 noetasuplem3 33938 noetasuplem4 33939 noetalem1 33944

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