Theorem nosupno 27878

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), 2o⟩}), (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 3475	. 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), 2o⟩}), (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 4492	. . . . . 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), 2o⟩}), (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), 2o⟩}))
4	3	adantr 485	. . . . 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), 2o⟩}), (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), 2o⟩}))
5		simprl 782	. . . . . . 7 ⊢ ((∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V)) → 𝐴 ⊆ No )
6		simpl 487	. . . . . . . . 9 ⊢ ((∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V)) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
7		nomaxmo 27873	. . . . . . . . . 10 ⊢ (𝐴 ⊆ No → ∃*𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
8	7	ad2antrl 740	. . . . . . . . 9 ⊢ ((∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V)) → ∃*𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
9		reu5 3370	. . . . . . . . 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 594	. . . . . . . 8 ⊢ ((∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V)) → ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
11		riotacl 7386	. . . . . . . 8 ⊢ (∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 → (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∈ 𝐴)
12	10, 11	syl 18	. . . . . . 7 ⊢ ((∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V)) → (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∈ 𝐴)
13	5, 12	sseldd 3937	. . . . . 6 ⊢ ((∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V)) → (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∈ No )
14		2on 8465	. . . . . . . . 9 ⊢ 2o ∈ On
15	14	elexi 3476	. . . . . . . 8 ⊢ 2o ∈ V
16	15	prid2 4728	. . . . . . 7 ⊢ 2o ∈ {1o, 2o}
17	16	noextend 27841	. . . . . 6 ⊢ ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∈ No → ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}) ∈ No )
18	13, 17	syl 18	. . . . 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), 2o⟩}) ∈ No )
19	4, 18	eqeltrd 2862	. . . 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), 2o⟩}), (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 4495	. . . . . 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), 2o⟩}), (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 485	. . . . 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), 2o⟩}), (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 6574	. . . . . . 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 6512	. . . . . . . . 9 ⊢ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)) ∈ V
25		eqid 2762	. . . . . . . . 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 6679	. . . . . . . 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 3931	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ⊆ No ∧ 𝑢 ∈ 𝐴) → 𝑢 ∈ No )
28		nodmon 27825	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ No → dom 𝑢 ∈ On)
29	27, 28	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ⊆ No ∧ 𝑢 ∈ 𝐴) → dom 𝑢 ∈ On)
30		onss 7782	. . . . . . . . . . . . . . . 16 ⊢ (dom 𝑢 ∈ On → dom 𝑢 ⊆ On)
31	29, 30	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ No ∧ 𝑢 ∈ 𝐴) → dom 𝑢 ⊆ On)
32	31	sseld 3935	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ No ∧ 𝑢 ∈ 𝐴) → (𝑦 ∈ dom 𝑢 → 𝑦 ∈ On))
33	32	adantrd 496	. . . . . . . . . . . . 13 ⊢ ((𝐴 ⊆ No ∧ 𝑢 ∈ 𝐴) → ((𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) → 𝑦 ∈ On))
34	33	rexlimdva 3165	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ No → (∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) → 𝑦 ∈ On))
35	34	abssdv 4020	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ No → {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ⊆ On)
36		simplr 780	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐴) → 𝑎 ∈ 𝑏)
37	29	adantlr 727	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐴) → dom 𝑢 ∈ On)
38		ontr1 6408	. . . . . . . . . . . . . . . . . . . 20 ⊢ (dom 𝑢 ∈ On → ((𝑎 ∈ 𝑏 ∧ 𝑏 ∈ dom 𝑢) → 𝑎 ∈ dom 𝑢))
39	37, 38	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐴) → ((𝑎 ∈ 𝑏 ∧ 𝑏 ∈ dom 𝑢) → 𝑎 ∈ dom 𝑢))
40	36, 39	mpand 707	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐴) → (𝑏 ∈ dom 𝑢 → 𝑎 ∈ dom 𝑢))
41	40	adantrd 496	. . . . . . . . . . . . . . . . 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 5971	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏) → ((𝑢 ↾ suc 𝑏) ↾ suc 𝑎) = ((𝑣 ↾ suc 𝑏) ↾ suc 𝑎))
43		onelon 6385	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((dom 𝑢 ∈ On ∧ 𝑏 ∈ dom 𝑢) → 𝑏 ∈ On)
44	37, 43	sylan 591	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐴 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐴) ∧ 𝑏 ∈ dom 𝑢) → 𝑏 ∈ On)
45		onsuc 7807	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑏 ∈ On → suc 𝑏 ∈ On)
46	44, 45	syl 18	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐴 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐴) ∧ 𝑏 ∈ dom 𝑢) → suc 𝑏 ∈ On)
47		simpllr 787	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐴 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐴) ∧ 𝑏 ∈ dom 𝑢) → 𝑎 ∈ 𝑏)
48		eloni 6370	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑏 ∈ On → Ord 𝑏)
49	44, 48	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐴 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐴) ∧ 𝑏 ∈ dom 𝑢) → Ord 𝑏)
50		ordsucelsuc 7816	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (Ord 𝑏 → (𝑎 ∈ 𝑏 ↔ suc 𝑎 ∈ suc 𝑏))
51	49, 50	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐴 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐴) ∧ 𝑏 ∈ dom 𝑢) → (𝑎 ∈ 𝑏 ↔ suc 𝑎 ∈ suc 𝑏))
52	47, 51	mpbid 235	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐴 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐴) ∧ 𝑏 ∈ dom 𝑢) → suc 𝑎 ∈ suc 𝑏)
53		onelss 6403	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (suc 𝑏 ∈ On → (suc 𝑎 ∈ suc 𝑏 → suc 𝑎 ⊆ suc 𝑏))
54	46, 52, 53	sylc 66	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐴 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐴) ∧ 𝑏 ∈ dom 𝑢) → suc 𝑎 ⊆ suc 𝑏)
55	54	resabs1d 6006	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐴 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐴) ∧ 𝑏 ∈ dom 𝑢) → ((𝑢 ↾ suc 𝑏) ↾ suc 𝑎) = (𝑢 ↾ suc 𝑎))
56	54	resabs1d 6006	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐴 ⊆ No ∧ 𝑎 ∈ 𝑏) ∧ 𝑢 ∈ 𝐴) ∧ 𝑏 ∈ dom 𝑢) → ((𝑣 ↾ suc 𝑏) ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎))
57	55, 56	eqeq12d 2778	. . . . . . . . . . . . . . . . . . . . 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	imbitrid 247	. . . . . . . . . . . . . . . . . . . 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 58	. . . . . . . . . . . . . . . . . . 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 3178	. . . . . . . . . . . . . . . . . 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 458	. . . . . . . . . . . . . . . . 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 521	. . . . . . . . . . . . . . . 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 3177	. . . . . . . . . . . . . . 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 458	. . . . . . . . . . . . . 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 3458	. . . . . . . . . . . . . . . 16 ⊢ 𝑏 ∈ V
66		eleq1w 2845	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑏 → (𝑦 ∈ dom 𝑢 ↔ 𝑏 ∈ dom 𝑢))
67		suceq 6429	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = 𝑏 → suc 𝑦 = suc 𝑏)
68	67	reseq2d 5977	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑏 → (𝑢 ↾ suc 𝑦) = (𝑢 ↾ suc 𝑏))
69	67	reseq2d 5977	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑏 → (𝑣 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑏))
70	68, 69	eqeq12d 2778	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑏 → ((𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦) ↔ (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏)))
71	70	imbi2d 343	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑏 → ((¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑏) = (𝑣 ↾ suc 𝑏))))
72	71	ralbidv 3187	. . . . . . . . . . . . . . . . . 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 643	. . . . . . . . . . . . . . . . 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 3188	. . . . . . . . . . . . . . . 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 3637	. . . . . . . . . . . . . . 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 634	. . . . . . . . . . . . . 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 3458	. . . . . . . . . . . . . . 15 ⊢ 𝑎 ∈ V
78		eleq1w 2845	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑎 → (𝑦 ∈ dom 𝑢 ↔ 𝑎 ∈ dom 𝑢))
79		suceq 6429	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑎 → suc 𝑦 = suc 𝑎)
80	79	reseq2d 5977	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑎 → (𝑢 ↾ suc 𝑦) = (𝑢 ↾ suc 𝑎))
81	79	reseq2d 5977	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑎 → (𝑣 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑎))
82	80, 81	eqeq12d 2778	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑎 → ((𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦) ↔ (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎)))
83	82	imbi2d 343	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑎 → ((¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑎) = (𝑣 ↾ suc 𝑎))))
84	83	ralbidv 3187	. . . . . . . . . . . . . . . . 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 643	. . . . . . . . . . . . . . . 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 3188	. . . . . . . . . . . . . . 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 3637	. . . . . . . . . . . . . 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 299	. . . . . . . . . . . . 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 1957	. . . . . . . . . . . 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 5219	. . . . . . . . . . . 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 237	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ No → Tr {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))})
92		dford5 7781	. . . . . . . . . . 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 594	. . . . . . . . . 10 ⊢ (𝐴 ⊆ No → Ord {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))})
94	93	adantr 485	. . . . . . . . 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 27852	. . . . . . . . . . . . . . 15 ⊢ bday : No –onto→On
96		fofun 6793	. . . . . . . . . . . . . . 15 ⊢ ( bday : No –onto→On → Fun bday )
97	95, 96	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ Fun bday
98		funimaexg 6622	. . . . . . . . . . . . . 14 ⊢ ((Fun bday ∧ 𝐴 ∈ V) → ( bday “ 𝐴) ∈ V)
99	97, 98	mpan 702	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ V → ( bday “ 𝐴) ∈ V)
100	99	uniexd 7742	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ V → ∪ ( bday “ 𝐴) ∈ V)
101	100	adantl 486	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V) → ∪ ( bday “ 𝐴) ∈ V)
102		simpl 487	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦))) → 𝑦 ∈ dom 𝑢)
103	102	reximi 3102	. . . . . . . . . . . . 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 4019	. . . . . . . . . . . 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 27823	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 ∈ No → ( bday ‘𝑢) = dom 𝑢)
106	27, 105	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ⊆ No ∧ 𝑢 ∈ 𝐴) → ( bday ‘𝑢) = dom 𝑢)
107		fofn 6794	. . . . . . . . . . . . . . . . . . . 20 ⊢ ( bday : No –onto→On → bday Fn No )
108	95, 107	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ bday Fn No
109		fnfvima 7231	. . . . . . . . . . . . . . . . . . 19 ⊢ (( bday Fn No ∧ 𝐴 ⊆ No ∧ 𝑢 ∈ 𝐴) → ( bday ‘𝑢) ∈ ( bday “ 𝐴))
110	108, 109	mp3an1 1476	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ⊆ No ∧ 𝑢 ∈ 𝐴) → ( bday ‘𝑢) ∈ ( bday “ 𝐴))
111	106, 110	eqeltrrd 2863	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ⊆ No ∧ 𝑢 ∈ 𝐴) → dom 𝑢 ∈ ( bday “ 𝐴))
112		elssuni 4903	. . . . . . . . . . . . . . . . 17 ⊢ (dom 𝑢 ∈ ( bday “ 𝐴) → dom 𝑢 ⊆ ∪ ( bday “ 𝐴))
113	111, 112	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ⊆ No ∧ 𝑢 ∈ 𝐴) → dom 𝑢 ⊆ ∪ ( bday “ 𝐴))
114	113	sseld 3935	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ No ∧ 𝑢 ∈ 𝐴) → (𝑦 ∈ dom 𝑢 → 𝑦 ∈ ∪ ( bday “ 𝐴)))
115	114	rexlimdva 3165	. . . . . . . . . . . . . 14 ⊢ (𝐴 ⊆ No → (∃𝑢 ∈ 𝐴 𝑦 ∈ dom 𝑢 → 𝑦 ∈ ∪ ( bday “ 𝐴)))
116	115	abssdv 4020	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ No → {𝑦 ∣ ∃𝑢 ∈ 𝐴 𝑦 ∈ dom 𝑢} ⊆ ∪ ( bday “ 𝐴))
117	116	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V) → {𝑦 ∣ ∃𝑢 ∈ 𝐴 𝑦 ∈ dom 𝑢} ⊆ ∪ ( bday “ 𝐴))
118	104, 117	sstrid 3947	. . . . . . . . . . 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 5294	. . . . . . . . . 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 6368	. . . . . . . . . 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 18	. . . . . . . . 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 260	. . . . . . . 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 2866	. . . . . . 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 486	. . . . . 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 5946	. . . . . . . 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 3458	. . . . . . . . . . . 12 ⊢ 𝑔 ∈ V
127		eleq1w 2845	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑔 → (𝑦 ∈ dom 𝑢 ↔ 𝑔 ∈ dom 𝑢))
128		suceq 6429	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑔 → suc 𝑦 = suc 𝑔)
129	128	reseq2d 5977	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑔 → (𝑢 ↾ suc 𝑦) = (𝑢 ↾ suc 𝑔))
130	128	reseq2d 5977	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑔 → (𝑣 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑔))
131	129, 130	eqeq12d 2778	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑔 → ((𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦) ↔ (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)))
132	131	imbi2d 343	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑔 → ((¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)) ↔ (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔))))
133	132	ralbidv 3187	. . . . . . . . . . . . . 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 643	. . . . . . . . . . . . 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 3188	. . . . . . . . . . . 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 3637	. . . . . . . . . . 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 2762	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢‘𝑔) = (𝑢‘𝑔)
138		fvex 6894	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢‘𝑔) ∈ V
139		eqeq2 2774	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = (𝑢‘𝑔) → ((𝑢‘𝑔) = 𝑥 ↔ (𝑢‘𝑔) = (𝑢‘𝑔)))
140	139	3anbi3d 1469	. . . . . . . . . . . . . . . . . . . 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 3564	. . . . . . . . . . . . . . . . . . 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 1478	. . . . . . . . . . . . . . . . . 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 3102	. . . . . . . . . . . . . . . . 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 3291	. . . . . . . . . . . . . . . . 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 221	. . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . 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 27875	. . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . 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 2596	. . . . . . . . . . . . . . 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 594	. . . . . . . . . . . . . 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 3458	. . . . . . . . . . . . . . 15 ⊢ 𝑧 ∈ V
152		eqeq2 2774	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑧 → ((𝑢‘𝑔) = 𝑥 ↔ (𝑢‘𝑔) = 𝑧))
153	152	3anbi3d 1469	. . . . . . . . . . . . . . . . 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 3188	. . . . . . . . . . . . . . . 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 6525	. . . . . . . . . . . . . . 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 702	. . . . . . . . . . . . . 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 18	. . . . . . . . . . . . 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 2769	. . . . . . . . . . . . 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 290	. . . . . . . . . . . 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 1241	. . . . . . . . . . . . . . 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 729	. . . . . . . . . . . . . . . . 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 27826	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ No → ran 𝑢 ⊆ {1o, 2o})
163	161, 162	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ⊆ No ∧ (𝑢 ∈ 𝐴 ∧ (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑧))) → ran 𝑢 ⊆ {1o, 2o})
164		nofun 27824	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 ∈ No → Fun 𝑢)
165	161, 164	syl 18	. . . . . . . . . . . . . . . . 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 1239	. . . . . . . . . . . . . . . . 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 7071	. . . . . . . . . . . . . . . . 17 ⊢ ((Fun 𝑢 ∧ 𝑔 ∈ dom 𝑢) → (𝑢‘𝑔) ∈ ran 𝑢)
168	165, 166, 167	syl2anc 595	. . . . . . . . . . . . . . . 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 3937	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ⊆ No ∧ (𝑢 ∈ 𝐴 ∧ (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑧))) → (𝑢‘𝑔) ∈ {1o, 2o})
170	160, 169	eqeltrrd 2863	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ No ∧ (𝑢 ∈ 𝐴 ∧ (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑧))) → 𝑧 ∈ {1o, 2o})
171	170	rexlimdvaa 3166	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ No → (∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑧) → 𝑧 ∈ {1o, 2o}))
172	171	adantr 485	. . . . . . . . . . . 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 ∈ {1o, 2o}))
173	159, 172	sylbird 263	. . . . . . . . . . 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 ∈ {1o, 2o}))
174	136, 173	sylan2b 605	. . . . . . . . . 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 ∈ {1o, 2o}))
175	174	rexlimdva 3165	. . . . . . . . 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 ∈ {1o, 2o}))
176	175	abssdv 4020	. . . . . . . 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))} ⊆ {1o, 2o})
177	125, 176	eqsstrid 3974	. . . . . . 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))) ⊆ {1o, 2o})
178	177	ad2antrl 740	. . . . . 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))) ⊆ {1o, 2o})
179		elno2 27829	. . . . . 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))) ⊆ {1o, 2o}))
180	23, 124, 178, 179	syl3anbrc 1361	. . . . 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 2862	. . . 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), 2o⟩}), (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 823	. . 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), 2o⟩}), (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 2866	. 2 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V) → 𝑆 ∈ No )
184	1, 183	sylan2 604	1 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ 𝑉) → 𝑆 ∈ No )

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1102  ∀wal 1567   = wceq 1569  ∃wex 1808   ∈ wcel 2142  ∃*wmo 2564  ∃!weu 2595  {cab 2740  ∀wral 3078  ∃wrex 3088  ∃!wreu 3366  ∃*wrmo 3367  Vcvv 3454   ∪ cun 3902   ⊆ wss 3904  ifcif 4486  {csn 4588  {cpr 4590  ⟨cop 4594  ∪ cuni 4871   class class class wbr 5108   ↦ cmpt 5191  Tr wtr 5217  dom cdm 5660  ran crn 5661   ↾ cres 5662   “ cima 5663  Ord word 6359  Oncon0 6360  suc csuc 6362  ℩cio 6490  Fun wfun 6530   Fn wfn 6531  –onto→wfo 6534  ‘cfv 6536  ℩crio 7368  1_oc1o 8444  2_oc2o 8445   No csur 27815   <s clts 27816   bday cbday 27817

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-ord 6363  df-on 6364  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fo 6542  df-fv 6544  df-riota 7369  df-1o 8451  df-2o 8452  df-no 27818  df-lts 27819  df-bday 27820

This theorem is used by:  nosupbday  27880  nosupres  27882  nosupbnd1lem1  27883  nosupbnd1lem2  27884  nosupbnd1lem3  27885  nosupbnd1lem4  27886  nosupbnd1lem5  27887  nosupbnd1lem6  27888  nosupbnd2  27891  nosupinfsep  27907  noetasuplem1  27908  noetasuplem2  27909  noetasuplem3  27910  noetasuplem4  27911  noetalem1  27916