noinfbnd2lem1 - Mathbox for Scott Fenton

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Theorem noinfbnd2lem1 33860

Description: Bounding law from below when a set of surreals has a minimum. (Contributed by Scott Fenton, 9-Aug-2024.)

**Assertion**
Ref	Expression
noinfbnd2lem1	⊢ (((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) → ¬ (𝑈 ∪ {⟨dom 𝑈, 1_o⟩}) <s (𝑍 ↾ suc dom 𝑈))

Distinct variable groups: 𝐵,𝑏 𝑈,𝑏 𝑍,𝑏 Allowed substitution hints: 𝐵(𝑦) 𝑈(𝑦) 𝑉(𝑦,𝑏) 𝑍(𝑦)

Proof of Theorem noinfbnd2lem1 Dummy variables 𝑞 𝑝 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 5074	. . 3 ⊢ (𝑏 = 𝑈 → (𝑍 <s 𝑏 ↔ 𝑍 <s 𝑈))
2		simp3 1136	. . 3 ⊢ (((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) → ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏)
3		simp1l 1195	. . 3 ⊢ (((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) → 𝑈 ∈ 𝐵)
4	1, 2, 3	rspcdva 3554	. 2 ⊢ (((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) → 𝑍 <s 𝑈)
5		simpl21 1249	. . . . . . . . . . 11 ⊢ ((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) → 𝐵 ⊆ No )
6		simpl1l 1222	. . . . . . . . . . 11 ⊢ ((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) → 𝑈 ∈ 𝐵)
7	5, 6	sseldd 3918	. . . . . . . . . 10 ⊢ ((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) → 𝑈 ∈ No )
8		nodmon 33780	. . . . . . . . . 10 ⊢ (𝑈 ∈ No → dom 𝑈 ∈ On)
9	7, 8	syl 17	. . . . . . . . 9 ⊢ ((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) → dom 𝑈 ∈ On)
10		onelon 6276	. . . . . . . . 9 ⊢ ((dom 𝑈 ∈ On ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ On)
11	9, 10	sylan 579	. . . . . . . 8 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ On)
12		simpr 484	. . . . . . . . . . . 12 ⊢ ((((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) ∧ 𝑞 ∈ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}) → 𝑞 ∈ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)})
13		simplr 765	. . . . . . . . . . . 12 ⊢ ((((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) ∧ 𝑞 ∈ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}) → ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈)
14	9	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → dom 𝑈 ∈ On)
15	14	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) ∧ 𝑞 ∈ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}) → dom 𝑈 ∈ On)
16		ontr1 6297	. . . . . . . . . . . . 13 ⊢ (dom 𝑈 ∈ On → ((𝑞 ∈ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → 𝑞 ∈ dom 𝑈))
17	15, 16	syl 17	. . . . . . . . . . . 12 ⊢ ((((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) ∧ 𝑞 ∈ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}) → ((𝑞 ∈ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → 𝑞 ∈ dom 𝑈))
18	12, 13, 17	mp2and 695	. . . . . . . . . . 11 ⊢ ((((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) ∧ 𝑞 ∈ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}) → 𝑞 ∈ dom 𝑈)
19	18	fvresd 6776	. . . . . . . . . 10 ⊢ ((((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) ∧ 𝑞 ∈ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}) → ((𝑍 ↾ dom 𝑈)‘𝑞) = (𝑍‘𝑞))
20		onelon 6276	. . . . . . . . . . . . 13 ⊢ ((∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ On ∧ 𝑞 ∈ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}) → 𝑞 ∈ On)
21	11, 20	sylan 579	. . . . . . . . . . . 12 ⊢ ((((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) ∧ 𝑞 ∈ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}) → 𝑞 ∈ On)
22		fveq2 6756	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑞 → (𝑈‘𝑥) = (𝑈‘𝑞))
23		fveq2 6756	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑞 → (𝑍‘𝑥) = (𝑍‘𝑞))
24	22, 23	neeq12d 3004	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑞 → ((𝑈‘𝑥) ≠ (𝑍‘𝑥) ↔ (𝑈‘𝑞) ≠ (𝑍‘𝑞)))
25	24	onnminsb 7626	. . . . . . . . . . . 12 ⊢ (𝑞 ∈ On → (𝑞 ∈ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} → ¬ (𝑈‘𝑞) ≠ (𝑍‘𝑞)))
26	21, 12, 25	sylc 65	. . . . . . . . . . 11 ⊢ ((((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) ∧ 𝑞 ∈ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}) → ¬ (𝑈‘𝑞) ≠ (𝑍‘𝑞))
27		df-ne 2943	. . . . . . . . . . . 12 ⊢ ((𝑈‘𝑞) ≠ (𝑍‘𝑞) ↔ ¬ (𝑈‘𝑞) = (𝑍‘𝑞))
28	27	con2bii 357	. . . . . . . . . . 11 ⊢ ((𝑈‘𝑞) = (𝑍‘𝑞) ↔ ¬ (𝑈‘𝑞) ≠ (𝑍‘𝑞))
29	26, 28	sylibr 233	. . . . . . . . . 10 ⊢ ((((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) ∧ 𝑞 ∈ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}) → (𝑈‘𝑞) = (𝑍‘𝑞))
30	19, 29	eqtr4d 2781	. . . . . . . . 9 ⊢ ((((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) ∧ 𝑞 ∈ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}) → ((𝑍 ↾ dom 𝑈)‘𝑞) = (𝑈‘𝑞))
31	30	ralrimiva 3107	. . . . . . . 8 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → ∀𝑞 ∈ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ((𝑍 ↾ dom 𝑈)‘𝑞) = (𝑈‘𝑞))
32		simpr 484	. . . . . . . . . 10 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈)
33	32	fvresd 6776	. . . . . . . . 9 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → ((𝑍 ↾ dom 𝑈)‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}) = (𝑍‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}))
34		simplr 765	. . . . . . . . . . 11 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → 𝑍 <s 𝑈)
35		simpl23 1251	. . . . . . . . . . . 12 ⊢ ((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) → 𝑍 ∈ No )
36	7	adantr 480	. . . . . . . . . . . 12 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → 𝑈 ∈ No )
37		sltval2 33786	. . . . . . . . . . . 12 ⊢ ((𝑍 ∈ No ∧ 𝑈 ∈ No ) → (𝑍 <s 𝑈 ↔ (𝑍‘∩ {𝑥 ∈ On ∣ (𝑍‘𝑥) ≠ (𝑈‘𝑥)}){⟨1_o, ∅⟩, ⟨1_o, 2_o⟩, ⟨∅, 2_o⟩} (𝑈‘∩ {𝑥 ∈ On ∣ (𝑍‘𝑥) ≠ (𝑈‘𝑥)})))
38	35, 36, 37	syl2an2r 681	. . . . . . . . . . 11 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → (𝑍 <s 𝑈 ↔ (𝑍‘∩ {𝑥 ∈ On ∣ (𝑍‘𝑥) ≠ (𝑈‘𝑥)}){⟨1_o, ∅⟩, ⟨1_o, 2_o⟩, ⟨∅, 2_o⟩} (𝑈‘∩ {𝑥 ∈ On ∣ (𝑍‘𝑥) ≠ (𝑈‘𝑥)})))
39	34, 38	mpbid 231	. . . . . . . . . 10 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → (𝑍‘∩ {𝑥 ∈ On ∣ (𝑍‘𝑥) ≠ (𝑈‘𝑥)}){⟨1_o, ∅⟩, ⟨1_o, 2_o⟩, ⟨∅, 2_o⟩} (𝑈‘∩ {𝑥 ∈ On ∣ (𝑍‘𝑥) ≠ (𝑈‘𝑥)}))
40		necom 2996	. . . . . . . . . . . . 13 ⊢ ((𝑈‘𝑥) ≠ (𝑍‘𝑥) ↔ (𝑍‘𝑥) ≠ (𝑈‘𝑥))
41	40	rabbii 3397	. . . . . . . . . . . 12 ⊢ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = {𝑥 ∈ On ∣ (𝑍‘𝑥) ≠ (𝑈‘𝑥)}
42	41	inteqi 4880	. . . . . . . . . . 11 ⊢ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = ∩ {𝑥 ∈ On ∣ (𝑍‘𝑥) ≠ (𝑈‘𝑥)}
43	42	fveq2i 6759	. . . . . . . . . 10 ⊢ (𝑍‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}) = (𝑍‘∩ {𝑥 ∈ On ∣ (𝑍‘𝑥) ≠ (𝑈‘𝑥)})
44	42	fveq2i 6759	. . . . . . . . . 10 ⊢ (𝑈‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}) = (𝑈‘∩ {𝑥 ∈ On ∣ (𝑍‘𝑥) ≠ (𝑈‘𝑥)})
45	39, 43, 44	3brtr4g 5104	. . . . . . . . 9 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → (𝑍‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}){⟨1_o, ∅⟩, ⟨1_o, 2_o⟩, ⟨∅, 2_o⟩} (𝑈‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}))
46	33, 45	eqbrtrd 5092	. . . . . . . 8 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → ((𝑍 ↾ dom 𝑈)‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}){⟨1_o, ∅⟩, ⟨1_o, 2_o⟩, ⟨∅, 2_o⟩} (𝑈‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}))
47		raleq 3333	. . . . . . . . . 10 ⊢ (𝑝 = ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} → (∀𝑞 ∈ 𝑝 ((𝑍 ↾ dom 𝑈)‘𝑞) = (𝑈‘𝑞) ↔ ∀𝑞 ∈ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ((𝑍 ↾ dom 𝑈)‘𝑞) = (𝑈‘𝑞)))
48		fveq2 6756	. . . . . . . . . . 11 ⊢ (𝑝 = ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} → ((𝑍 ↾ dom 𝑈)‘𝑝) = ((𝑍 ↾ dom 𝑈)‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}))
49		fveq2 6756	. . . . . . . . . . 11 ⊢ (𝑝 = ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} → (𝑈‘𝑝) = (𝑈‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}))
50	48, 49	breq12d 5083	. . . . . . . . . 10 ⊢ (𝑝 = ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} → (((𝑍 ↾ dom 𝑈)‘𝑝){⟨1_o, ∅⟩, ⟨1_o, 2_o⟩, ⟨∅, 2_o⟩} (𝑈‘𝑝) ↔ ((𝑍 ↾ dom 𝑈)‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}){⟨1_o, ∅⟩, ⟨1_o, 2_o⟩, ⟨∅, 2_o⟩} (𝑈‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)})))
51	47, 50	anbi12d 630	. . . . . . . . 9 ⊢ (𝑝 = ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} → ((∀𝑞 ∈ 𝑝 ((𝑍 ↾ dom 𝑈)‘𝑞) = (𝑈‘𝑞) ∧ ((𝑍 ↾ dom 𝑈)‘𝑝){⟨1_o, ∅⟩, ⟨1_o, 2_o⟩, ⟨∅, 2_o⟩} (𝑈‘𝑝)) ↔ (∀𝑞 ∈ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ((𝑍 ↾ dom 𝑈)‘𝑞) = (𝑈‘𝑞) ∧ ((𝑍 ↾ dom 𝑈)‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}){⟨1_o, ∅⟩, ⟨1_o, 2_o⟩, ⟨∅, 2_o⟩} (𝑈‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}))))
52	51	rspcev 3552	. . . . . . . 8 ⊢ ((∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ On ∧ (∀𝑞 ∈ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ((𝑍 ↾ dom 𝑈)‘𝑞) = (𝑈‘𝑞) ∧ ((𝑍 ↾ dom 𝑈)‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}){⟨1_o, ∅⟩, ⟨1_o, 2_o⟩, ⟨∅, 2_o⟩} (𝑈‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}))) → ∃𝑝 ∈ On (∀𝑞 ∈ 𝑝 ((𝑍 ↾ dom 𝑈)‘𝑞) = (𝑈‘𝑞) ∧ ((𝑍 ↾ dom 𝑈)‘𝑝){⟨1_o, ∅⟩, ⟨1_o, 2_o⟩, ⟨∅, 2_o⟩} (𝑈‘𝑝)))
53	11, 31, 46, 52	syl12anc 833	. . . . . . 7 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → ∃𝑝 ∈ On (∀𝑞 ∈ 𝑝 ((𝑍 ↾ dom 𝑈)‘𝑞) = (𝑈‘𝑞) ∧ ((𝑍 ↾ dom 𝑈)‘𝑝){⟨1_o, ∅⟩, ⟨1_o, 2_o⟩, ⟨∅, 2_o⟩} (𝑈‘𝑝)))
54		noreson 33790	. . . . . . . . 9 ⊢ ((𝑍 ∈ No ∧ dom 𝑈 ∈ On) → (𝑍 ↾ dom 𝑈) ∈ No )
55	35, 9, 54	syl2anc 583	. . . . . . . 8 ⊢ ((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) → (𝑍 ↾ dom 𝑈) ∈ No )
56		sltval 33777	. . . . . . . 8 ⊢ (((𝑍 ↾ dom 𝑈) ∈ No ∧ 𝑈 ∈ No ) → ((𝑍 ↾ dom 𝑈) <s 𝑈 ↔ ∃𝑝 ∈ On (∀𝑞 ∈ 𝑝 ((𝑍 ↾ dom 𝑈)‘𝑞) = (𝑈‘𝑞) ∧ ((𝑍 ↾ dom 𝑈)‘𝑝){⟨1_o, ∅⟩, ⟨1_o, 2_o⟩, ⟨∅, 2_o⟩} (𝑈‘𝑝))))
57	55, 36, 56	syl2an2r 681	. . . . . . 7 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → ((𝑍 ↾ dom 𝑈) <s 𝑈 ↔ ∃𝑝 ∈ On (∀𝑞 ∈ 𝑝 ((𝑍 ↾ dom 𝑈)‘𝑞) = (𝑈‘𝑞) ∧ ((𝑍 ↾ dom 𝑈)‘𝑝){⟨1_o, ∅⟩, ⟨1_o, 2_o⟩, ⟨∅, 2_o⟩} (𝑈‘𝑝))))
58	53, 57	mpbird 256	. . . . . 6 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → (𝑍 ↾ dom 𝑈) <s 𝑈)
59		sssucid 6328	. . . . . . 7 ⊢ dom 𝑈 ⊆ suc dom 𝑈
60		resabs1 5910	. . . . . . 7 ⊢ (dom 𝑈 ⊆ suc dom 𝑈 → ((𝑍 ↾ suc dom 𝑈) ↾ dom 𝑈) = (𝑍 ↾ dom 𝑈))
61	59, 60	mp1i 13	. . . . . 6 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → ((𝑍 ↾ suc dom 𝑈) ↾ dom 𝑈) = (𝑍 ↾ dom 𝑈))
62		resundir 5895	. . . . . . 7 ⊢ ((𝑈 ∪ {⟨dom 𝑈, 1_o⟩}) ↾ dom 𝑈) = ((𝑈 ↾ dom 𝑈) ∪ ({⟨dom 𝑈, 1_o⟩} ↾ dom 𝑈))
63		nofun 33779	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ No → Fun 𝑈)
64	7, 63	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) → Fun 𝑈)
65		funrel 6435	. . . . . . . . . . . 12 ⊢ (Fun 𝑈 → Rel 𝑈)
66	64, 65	syl 17	. . . . . . . . . . 11 ⊢ ((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) → Rel 𝑈)
67	66	adantr 480	. . . . . . . . . 10 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → Rel 𝑈)
68		resdm 5925	. . . . . . . . . 10 ⊢ (Rel 𝑈 → (𝑈 ↾ dom 𝑈) = 𝑈)
69	67, 68	syl 17	. . . . . . . . 9 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → (𝑈 ↾ dom 𝑈) = 𝑈)
70		nodmord 33783	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ No → Ord dom 𝑈)
71	7, 70	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) → Ord dom 𝑈)
72	71	adantr 480	. . . . . . . . . . 11 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → Ord dom 𝑈)
73		ordirr 6269	. . . . . . . . . . 11 ⊢ (Ord dom 𝑈 → ¬ dom 𝑈 ∈ dom 𝑈)
74	72, 73	syl 17	. . . . . . . . . 10 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → ¬ dom 𝑈 ∈ dom 𝑈)
75		1oex 8280	. . . . . . . . . . 11 ⊢ 1_o ∈ V
76	75	snres0 33577	. . . . . . . . . 10 ⊢ (({⟨dom 𝑈, 1_o⟩} ↾ dom 𝑈) = ∅ ↔ ¬ dom 𝑈 ∈ dom 𝑈)
77	74, 76	sylibr 233	. . . . . . . . 9 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → ({⟨dom 𝑈, 1_o⟩} ↾ dom 𝑈) = ∅)
78	69, 77	uneq12d 4094	. . . . . . . 8 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → ((𝑈 ↾ dom 𝑈) ∪ ({⟨dom 𝑈, 1_o⟩} ↾ dom 𝑈)) = (𝑈 ∪ ∅))
79		un0 4321	. . . . . . . 8 ⊢ (𝑈 ∪ ∅) = 𝑈
80	78, 79	eqtrdi 2795	. . . . . . 7 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → ((𝑈 ↾ dom 𝑈) ∪ ({⟨dom 𝑈, 1_o⟩} ↾ dom 𝑈)) = 𝑈)
81	62, 80	syl5eq 2791	. . . . . 6 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → ((𝑈 ∪ {⟨dom 𝑈, 1_o⟩}) ↾ dom 𝑈) = 𝑈)
82	58, 61, 81	3brtr4d 5102	. . . . 5 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → ((𝑍 ↾ suc dom 𝑈) ↾ dom 𝑈) <s ((𝑈 ∪ {⟨dom 𝑈, 1_o⟩}) ↾ dom 𝑈))
83		sucelon 7639	. . . . . . . . 9 ⊢ (dom 𝑈 ∈ On ↔ suc dom 𝑈 ∈ On)
84	9, 83	sylib 217	. . . . . . . 8 ⊢ ((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) → suc dom 𝑈 ∈ On)
85		noreson 33790	. . . . . . . 8 ⊢ ((𝑍 ∈ No ∧ suc dom 𝑈 ∈ On) → (𝑍 ↾ suc dom 𝑈) ∈ No )
86	35, 84, 85	syl2anc 583	. . . . . . 7 ⊢ ((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) → (𝑍 ↾ suc dom 𝑈) ∈ No )
87	86	adantr 480	. . . . . 6 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → (𝑍 ↾ suc dom 𝑈) ∈ No )
88	75	prid1 4695	. . . . . . . . 9 ⊢ 1_o ∈ {1_o, 2_o}
89	88	noextend 33796	. . . . . . . 8 ⊢ (𝑈 ∈ No → (𝑈 ∪ {⟨dom 𝑈, 1_o⟩}) ∈ No )
90	7, 89	syl 17	. . . . . . 7 ⊢ ((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) → (𝑈 ∪ {⟨dom 𝑈, 1_o⟩}) ∈ No )
91	90	adantr 480	. . . . . 6 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → (𝑈 ∪ {⟨dom 𝑈, 1_o⟩}) ∈ No )
92		sltres 33792	. . . . . 6 ⊢ (((𝑍 ↾ suc dom 𝑈) ∈ No ∧ (𝑈 ∪ {⟨dom 𝑈, 1_o⟩}) ∈ No ∧ dom 𝑈 ∈ On) → (((𝑍 ↾ suc dom 𝑈) ↾ dom 𝑈) <s ((𝑈 ∪ {⟨dom 𝑈, 1_o⟩}) ↾ dom 𝑈) → (𝑍 ↾ suc dom 𝑈) <s (𝑈 ∪ {⟨dom 𝑈, 1_o⟩})))
93	87, 91, 14, 92	syl3anc 1369	. . . . 5 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → (((𝑍 ↾ suc dom 𝑈) ↾ dom 𝑈) <s ((𝑈 ∪ {⟨dom 𝑈, 1_o⟩}) ↾ dom 𝑈) → (𝑍 ↾ suc dom 𝑈) <s (𝑈 ∪ {⟨dom 𝑈, 1_o⟩})))
94	82, 93	mpd 15	. . . 4 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → (𝑍 ↾ suc dom 𝑈) <s (𝑈 ∪ {⟨dom 𝑈, 1_o⟩}))
95		sltso 33806	. . . . . 6 ⊢ <s Or No
96		soasym 5525	. . . . . 6 ⊢ (( <s Or No ∧ ((𝑍 ↾ suc dom 𝑈) ∈ No ∧ (𝑈 ∪ {⟨dom 𝑈, 1_o⟩}) ∈ No )) → ((𝑍 ↾ suc dom 𝑈) <s (𝑈 ∪ {⟨dom 𝑈, 1_o⟩}) → ¬ (𝑈 ∪ {⟨dom 𝑈, 1_o⟩}) <s (𝑍 ↾ suc dom 𝑈)))
97	95, 96	mpan 686	. . . . 5 ⊢ (((𝑍 ↾ suc dom 𝑈) ∈ No ∧ (𝑈 ∪ {⟨dom 𝑈, 1_o⟩}) ∈ No ) → ((𝑍 ↾ suc dom 𝑈) <s (𝑈 ∪ {⟨dom 𝑈, 1_o⟩}) → ¬ (𝑈 ∪ {⟨dom 𝑈, 1_o⟩}) <s (𝑍 ↾ suc dom 𝑈)))
98	86, 91, 97	syl2an2r 681	. . . 4 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → ((𝑍 ↾ suc dom 𝑈) <s (𝑈 ∪ {⟨dom 𝑈, 1_o⟩}) → ¬ (𝑈 ∪ {⟨dom 𝑈, 1_o⟩}) <s (𝑍 ↾ suc dom 𝑈)))
99	94, 98	mpd 15	. . 3 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → ¬ (𝑈 ∪ {⟨dom 𝑈, 1_o⟩}) <s (𝑍 ↾ suc dom 𝑈))
100		sonr 5517	. . . . . 6 ⊢ (( <s Or No ∧ (𝑈 ∪ {⟨dom 𝑈, 1_o⟩}) ∈ No ) → ¬ (𝑈 ∪ {⟨dom 𝑈, 1_o⟩}) <s (𝑈 ∪ {⟨dom 𝑈, 1_o⟩}))
101	95, 90, 100	sylancr 586	. . . . 5 ⊢ ((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) → ¬ (𝑈 ∪ {⟨dom 𝑈, 1_o⟩}) <s (𝑈 ∪ {⟨dom 𝑈, 1_o⟩}))
102	101	adantr 480	. . . 4 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → ¬ (𝑈 ∪ {⟨dom 𝑈, 1_o⟩}) <s (𝑈 ∪ {⟨dom 𝑈, 1_o⟩}))
103		df-suc 6257	. . . . . . . 8 ⊢ suc dom 𝑈 = (dom 𝑈 ∪ {dom 𝑈})
104	103	reseq2i 5877	. . . . . . 7 ⊢ (𝑍 ↾ suc dom 𝑈) = (𝑍 ↾ (dom 𝑈 ∪ {dom 𝑈}))
105		resundi 5894	. . . . . . 7 ⊢ (𝑍 ↾ (dom 𝑈 ∪ {dom 𝑈})) = ((𝑍 ↾ dom 𝑈) ∪ (𝑍 ↾ {dom 𝑈}))
106	104, 105	eqtri 2766	. . . . . 6 ⊢ (𝑍 ↾ suc dom 𝑈) = ((𝑍 ↾ dom 𝑈) ∪ (𝑍 ↾ {dom 𝑈}))
107		dmres 5902	. . . . . . . . 9 ⊢ dom (𝑍 ↾ dom 𝑈) = (dom 𝑈 ∩ dom 𝑍)
108	42	eqeq1i 2743	. . . . . . . . . . . . 13 ⊢ (∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈 ↔ ∩ {𝑥 ∈ On ∣ (𝑍‘𝑥) ≠ (𝑈‘𝑥)} = dom 𝑈)
109	108	biimpi 215	. . . . . . . . . . . 12 ⊢ (∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈 → ∩ {𝑥 ∈ On ∣ (𝑍‘𝑥) ≠ (𝑈‘𝑥)} = dom 𝑈)
110	109	adantl 481	. . . . . . . . . . 11 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → ∩ {𝑥 ∈ On ∣ (𝑍‘𝑥) ≠ (𝑈‘𝑥)} = dom 𝑈)
111	35	adantr 480	. . . . . . . . . . . 12 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → 𝑍 ∈ No )
112	7	adantr 480	. . . . . . . . . . . 12 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → 𝑈 ∈ No )
113		simp23 1206	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) → 𝑍 ∈ No )
114		sonr 5517	. . . . . . . . . . . . . . . . . 18 ⊢ (( <s Or No ∧ 𝑍 ∈ No ) → ¬ 𝑍 <s 𝑍)
115	95, 113, 114	sylancr 586	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) → ¬ 𝑍 <s 𝑍)
116		breq2 5074	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑈 = 𝑍 → (𝑍 <s 𝑈 ↔ 𝑍 <s 𝑍))
117	116	notbid 317	. . . . . . . . . . . . . . . . 17 ⊢ (𝑈 = 𝑍 → (¬ 𝑍 <s 𝑈 ↔ ¬ 𝑍 <s 𝑍))
118	115, 117	syl5ibrcom 246	. . . . . . . . . . . . . . . 16 ⊢ (((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) → (𝑈 = 𝑍 → ¬ 𝑍 <s 𝑈))
119	118	necon2ad 2957	. . . . . . . . . . . . . . 15 ⊢ (((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) → (𝑍 <s 𝑈 → 𝑈 ≠ 𝑍))
120	119	imp 406	. . . . . . . . . . . . . 14 ⊢ ((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) → 𝑈 ≠ 𝑍)
121	120	necomd 2998	. . . . . . . . . . . . 13 ⊢ ((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) → 𝑍 ≠ 𝑈)
122	121	adantr 480	. . . . . . . . . . . 12 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → 𝑍 ≠ 𝑈)
123		nosepssdm 33816	. . . . . . . . . . . 12 ⊢ ((𝑍 ∈ No ∧ 𝑈 ∈ No ∧ 𝑍 ≠ 𝑈) → ∩ {𝑥 ∈ On ∣ (𝑍‘𝑥) ≠ (𝑈‘𝑥)} ⊆ dom 𝑍)
124	111, 112, 122, 123	syl3anc 1369	. . . . . . . . . . 11 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → ∩ {𝑥 ∈ On ∣ (𝑍‘𝑥) ≠ (𝑈‘𝑥)} ⊆ dom 𝑍)
125	110, 124	eqsstrrd 3956	. . . . . . . . . 10 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → dom 𝑈 ⊆ dom 𝑍)
126		df-ss 3900	. . . . . . . . . 10 ⊢ (dom 𝑈 ⊆ dom 𝑍 ↔ (dom 𝑈 ∩ dom 𝑍) = dom 𝑈)
127	125, 126	sylib 217	. . . . . . . . 9 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → (dom 𝑈 ∩ dom 𝑍) = dom 𝑈)
128	107, 127	syl5eq 2791	. . . . . . . 8 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → dom (𝑍 ↾ dom 𝑈) = dom 𝑈)
129	128	eleq2d 2824	. . . . . . . . . 10 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → (𝑞 ∈ dom (𝑍 ↾ dom 𝑈) ↔ 𝑞 ∈ dom 𝑈))
130		simpr 484	. . . . . . . . . . . . 13 ⊢ ((((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) ∧ 𝑞 ∈ dom 𝑈) → 𝑞 ∈ dom 𝑈)
131	130	fvresd 6776	. . . . . . . . . . . 12 ⊢ ((((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) ∧ 𝑞 ∈ dom 𝑈) → ((𝑍 ↾ dom 𝑈)‘𝑞) = (𝑍‘𝑞))
132	112, 8	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → dom 𝑈 ∈ On)
133		onelon 6276	. . . . . . . . . . . . . . 15 ⊢ ((dom 𝑈 ∈ On ∧ 𝑞 ∈ dom 𝑈) → 𝑞 ∈ On)
134	132, 133	sylan 579	. . . . . . . . . . . . . 14 ⊢ ((((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) ∧ 𝑞 ∈ dom 𝑈) → 𝑞 ∈ On)
135		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈)
136	135	eleq2d 2824	. . . . . . . . . . . . . . 15 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → (𝑞 ∈ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ↔ 𝑞 ∈ dom 𝑈))
137	136	biimpar 477	. . . . . . . . . . . . . 14 ⊢ ((((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) ∧ 𝑞 ∈ dom 𝑈) → 𝑞 ∈ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)})
138	134, 137, 25	sylc 65	. . . . . . . . . . . . 13 ⊢ ((((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) ∧ 𝑞 ∈ dom 𝑈) → ¬ (𝑈‘𝑞) ≠ (𝑍‘𝑞))
139		nesym 2999	. . . . . . . . . . . . . 14 ⊢ ((𝑈‘𝑞) ≠ (𝑍‘𝑞) ↔ ¬ (𝑍‘𝑞) = (𝑈‘𝑞))
140	139	con2bii 357	. . . . . . . . . . . . 13 ⊢ ((𝑍‘𝑞) = (𝑈‘𝑞) ↔ ¬ (𝑈‘𝑞) ≠ (𝑍‘𝑞))
141	138, 140	sylibr 233	. . . . . . . . . . . 12 ⊢ ((((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) ∧ 𝑞 ∈ dom 𝑈) → (𝑍‘𝑞) = (𝑈‘𝑞))
142	131, 141	eqtrd 2778	. . . . . . . . . . 11 ⊢ ((((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) ∧ 𝑞 ∈ dom 𝑈) → ((𝑍 ↾ dom 𝑈)‘𝑞) = (𝑈‘𝑞))
143	142	ex 412	. . . . . . . . . 10 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → (𝑞 ∈ dom 𝑈 → ((𝑍 ↾ dom 𝑈)‘𝑞) = (𝑈‘𝑞)))
144	129, 143	sylbid 239	. . . . . . . . 9 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → (𝑞 ∈ dom (𝑍 ↾ dom 𝑈) → ((𝑍 ↾ dom 𝑈)‘𝑞) = (𝑈‘𝑞)))
145	144	ralrimiv 3106	. . . . . . . 8 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → ∀𝑞 ∈ dom (𝑍 ↾ dom 𝑈)((𝑍 ↾ dom 𝑈)‘𝑞) = (𝑈‘𝑞))
146		nofun 33779	. . . . . . . . . . 11 ⊢ (𝑍 ∈ No → Fun 𝑍)
147	111, 146	syl 17	. . . . . . . . . 10 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → Fun 𝑍)
148	147	funresd 6461	. . . . . . . . 9 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → Fun (𝑍 ↾ dom 𝑈))
149	64	adantr 480	. . . . . . . . 9 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → Fun 𝑈)
150		eqfunfv 6896	. . . . . . . . 9 ⊢ ((Fun (𝑍 ↾ dom 𝑈) ∧ Fun 𝑈) → ((𝑍 ↾ dom 𝑈) = 𝑈 ↔ (dom (𝑍 ↾ dom 𝑈) = dom 𝑈 ∧ ∀𝑞 ∈ dom (𝑍 ↾ dom 𝑈)((𝑍 ↾ dom 𝑈)‘𝑞) = (𝑈‘𝑞))))
151	148, 149, 150	syl2anc 583	. . . . . . . 8 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → ((𝑍 ↾ dom 𝑈) = 𝑈 ↔ (dom (𝑍 ↾ dom 𝑈) = dom 𝑈 ∧ ∀𝑞 ∈ dom (𝑍 ↾ dom 𝑈)((𝑍 ↾ dom 𝑈)‘𝑞) = (𝑈‘𝑞))))
152	128, 145, 151	mpbir2and 709	. . . . . . 7 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → (𝑍 ↾ dom 𝑈) = 𝑈)
153	35, 146	syl 17	. . . . . . . . . 10 ⊢ ((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) → Fun 𝑍)
154	153	funfnd 6449	. . . . . . . . 9 ⊢ ((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) → 𝑍 Fn dom 𝑍)
155	112, 70	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → Ord dom 𝑈)
156	155, 73	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → ¬ dom 𝑈 ∈ dom 𝑈)
157		ndmfv 6786	. . . . . . . . . . . . . . . 16 ⊢ (¬ dom 𝑈 ∈ dom 𝑈 → (𝑈‘dom 𝑈) = ∅)
158		2on0 8276	. . . . . . . . . . . . . . . . . . 19 ⊢ 2_o ≠ ∅
159	158	necomi 2997	. . . . . . . . . . . . . . . . . 18 ⊢ ∅ ≠ 2_o
160		neeq1 3005	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑈‘dom 𝑈) = ∅ → ((𝑈‘dom 𝑈) ≠ 2_o ↔ ∅ ≠ 2_o))
161	159, 160	mpbiri 257	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑈‘dom 𝑈) = ∅ → (𝑈‘dom 𝑈) ≠ 2_o)
162	161	neneqd 2947	. . . . . . . . . . . . . . . 16 ⊢ ((𝑈‘dom 𝑈) = ∅ → ¬ (𝑈‘dom 𝑈) = 2_o)
163	157, 162	syl 17	. . . . . . . . . . . . . . 15 ⊢ (¬ dom 𝑈 ∈ dom 𝑈 → ¬ (𝑈‘dom 𝑈) = 2_o)
164	156, 163	syl 17	. . . . . . . . . . . . . 14 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → ¬ (𝑈‘dom 𝑈) = 2_o)
165	164	intnand 488	. . . . . . . . . . . . 13 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → ¬ ((𝑍‘dom 𝑈) = ∅ ∧ (𝑈‘dom 𝑈) = 2_o))
166		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) → 𝑍 <s 𝑈)
167	35, 7, 37	syl2anc 583	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) → (𝑍 <s 𝑈 ↔ (𝑍‘∩ {𝑥 ∈ On ∣ (𝑍‘𝑥) ≠ (𝑈‘𝑥)}){⟨1_o, ∅⟩, ⟨1_o, 2_o⟩, ⟨∅, 2_o⟩} (𝑈‘∩ {𝑥 ∈ On ∣ (𝑍‘𝑥) ≠ (𝑈‘𝑥)})))
168	166, 167	mpbid 231	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) → (𝑍‘∩ {𝑥 ∈ On ∣ (𝑍‘𝑥) ≠ (𝑈‘𝑥)}){⟨1_o, ∅⟩, ⟨1_o, 2_o⟩, ⟨∅, 2_o⟩} (𝑈‘∩ {𝑥 ∈ On ∣ (𝑍‘𝑥) ≠ (𝑈‘𝑥)}))
169	168	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → (𝑍‘∩ {𝑥 ∈ On ∣ (𝑍‘𝑥) ≠ (𝑈‘𝑥)}){⟨1_o, ∅⟩, ⟨1_o, 2_o⟩, ⟨∅, 2_o⟩} (𝑈‘∩ {𝑥 ∈ On ∣ (𝑍‘𝑥) ≠ (𝑈‘𝑥)}))
170	110	fveq2d 6760	. . . . . . . . . . . . . . 15 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → (𝑍‘∩ {𝑥 ∈ On ∣ (𝑍‘𝑥) ≠ (𝑈‘𝑥)}) = (𝑍‘dom 𝑈))
171	110	fveq2d 6760	. . . . . . . . . . . . . . 15 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → (𝑈‘∩ {𝑥 ∈ On ∣ (𝑍‘𝑥) ≠ (𝑈‘𝑥)}) = (𝑈‘dom 𝑈))
172	169, 170, 171	3brtr3d 5101	. . . . . . . . . . . . . 14 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → (𝑍‘dom 𝑈){⟨1_o, ∅⟩, ⟨1_o, 2_o⟩, ⟨∅, 2_o⟩} (𝑈‘dom 𝑈))
173		fvex 6769	. . . . . . . . . . . . . . 15 ⊢ (𝑍‘dom 𝑈) ∈ V
174		fvex 6769	. . . . . . . . . . . . . . 15 ⊢ (𝑈‘dom 𝑈) ∈ V
175	173, 174	brtp 33623	. . . . . . . . . . . . . 14 ⊢ ((𝑍‘dom 𝑈){⟨1_o, ∅⟩, ⟨1_o, 2_o⟩, ⟨∅, 2_o⟩} (𝑈‘dom 𝑈) ↔ (((𝑍‘dom 𝑈) = 1_o ∧ (𝑈‘dom 𝑈) = ∅) ∨ ((𝑍‘dom 𝑈) = 1_o ∧ (𝑈‘dom 𝑈) = 2_o) ∨ ((𝑍‘dom 𝑈) = ∅ ∧ (𝑈‘dom 𝑈) = 2_o)))
176	172, 175	sylib 217	. . . . . . . . . . . . 13 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → (((𝑍‘dom 𝑈) = 1_o ∧ (𝑈‘dom 𝑈) = ∅) ∨ ((𝑍‘dom 𝑈) = 1_o ∧ (𝑈‘dom 𝑈) = 2_o) ∨ ((𝑍‘dom 𝑈) = ∅ ∧ (𝑈‘dom 𝑈) = 2_o)))
177		3orel3 33557	. . . . . . . . . . . . 13 ⊢ (¬ ((𝑍‘dom 𝑈) = ∅ ∧ (𝑈‘dom 𝑈) = 2_o) → ((((𝑍‘dom 𝑈) = 1_o ∧ (𝑈‘dom 𝑈) = ∅) ∨ ((𝑍‘dom 𝑈) = 1_o ∧ (𝑈‘dom 𝑈) = 2_o) ∨ ((𝑍‘dom 𝑈) = ∅ ∧ (𝑈‘dom 𝑈) = 2_o)) → (((𝑍‘dom 𝑈) = 1_o ∧ (𝑈‘dom 𝑈) = ∅) ∨ ((𝑍‘dom 𝑈) = 1_o ∧ (𝑈‘dom 𝑈) = 2_o))))
178	165, 176, 177	sylc 65	. . . . . . . . . . . 12 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → (((𝑍‘dom 𝑈) = 1_o ∧ (𝑈‘dom 𝑈) = ∅) ∨ ((𝑍‘dom 𝑈) = 1_o ∧ (𝑈‘dom 𝑈) = 2_o)))
179		andi 1004	. . . . . . . . . . . 12 ⊢ (((𝑍‘dom 𝑈) = 1_o ∧ ((𝑈‘dom 𝑈) = ∅ ∨ (𝑈‘dom 𝑈) = 2_o)) ↔ (((𝑍‘dom 𝑈) = 1_o ∧ (𝑈‘dom 𝑈) = ∅) ∨ ((𝑍‘dom 𝑈) = 1_o ∧ (𝑈‘dom 𝑈) = 2_o)))
180	178, 179	sylibr 233	. . . . . . . . . . 11 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → ((𝑍‘dom 𝑈) = 1_o ∧ ((𝑈‘dom 𝑈) = ∅ ∨ (𝑈‘dom 𝑈) = 2_o)))
181	180	simpld 494	. . . . . . . . . 10 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → (𝑍‘dom 𝑈) = 1_o)
182		ndmfv 6786	. . . . . . . . . . . 12 ⊢ (¬ dom 𝑈 ∈ dom 𝑍 → (𝑍‘dom 𝑈) = ∅)
183		1n0 8286	. . . . . . . . . . . . . . 15 ⊢ 1_o ≠ ∅
184	183	necomi 2997	. . . . . . . . . . . . . 14 ⊢ ∅ ≠ 1_o
185		neeq1 3005	. . . . . . . . . . . . . 14 ⊢ ((𝑍‘dom 𝑈) = ∅ → ((𝑍‘dom 𝑈) ≠ 1_o ↔ ∅ ≠ 1_o))
186	184, 185	mpbiri 257	. . . . . . . . . . . . 13 ⊢ ((𝑍‘dom 𝑈) = ∅ → (𝑍‘dom 𝑈) ≠ 1_o)
187	186	neneqd 2947	. . . . . . . . . . . 12 ⊢ ((𝑍‘dom 𝑈) = ∅ → ¬ (𝑍‘dom 𝑈) = 1_o)
188	182, 187	syl 17	. . . . . . . . . . 11 ⊢ (¬ dom 𝑈 ∈ dom 𝑍 → ¬ (𝑍‘dom 𝑈) = 1_o)
189	188	con4i 114	. . . . . . . . . 10 ⊢ ((𝑍‘dom 𝑈) = 1_o → dom 𝑈 ∈ dom 𝑍)
190	181, 189	syl 17	. . . . . . . . 9 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → dom 𝑈 ∈ dom 𝑍)
191		fnressn 7012	. . . . . . . . 9 ⊢ ((𝑍 Fn dom 𝑍 ∧ dom 𝑈 ∈ dom 𝑍) → (𝑍 ↾ {dom 𝑈}) = {⟨dom 𝑈, (𝑍‘dom 𝑈)⟩})
192	154, 190, 191	syl2an2r 681	. . . . . . . 8 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → (𝑍 ↾ {dom 𝑈}) = {⟨dom 𝑈, (𝑍‘dom 𝑈)⟩})
193	181	opeq2d 4808	. . . . . . . . 9 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → ⟨dom 𝑈, (𝑍‘dom 𝑈)⟩ = ⟨dom 𝑈, 1_o⟩)
194	193	sneqd 4570	. . . . . . . 8 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → {⟨dom 𝑈, (𝑍‘dom 𝑈)⟩} = {⟨dom 𝑈, 1_o⟩})
195	192, 194	eqtrd 2778	. . . . . . 7 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → (𝑍 ↾ {dom 𝑈}) = {⟨dom 𝑈, 1_o⟩})
196	152, 195	uneq12d 4094	. . . . . 6 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → ((𝑍 ↾ dom 𝑈) ∪ (𝑍 ↾ {dom 𝑈})) = (𝑈 ∪ {⟨dom 𝑈, 1_o⟩}))
197	106, 196	syl5eq 2791	. . . . 5 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → (𝑍 ↾ suc dom 𝑈) = (𝑈 ∪ {⟨dom 𝑈, 1_o⟩}))
198	197	breq2d 5082	. . . 4 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → ((𝑈 ∪ {⟨dom 𝑈, 1_o⟩}) <s (𝑍 ↾ suc dom 𝑈) ↔ (𝑈 ∪ {⟨dom 𝑈, 1_o⟩}) <s (𝑈 ∪ {⟨dom 𝑈, 1_o⟩})))
199	102, 198	mtbird 324	. . 3 ⊢ (((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → ¬ (𝑈 ∪ {⟨dom 𝑈, 1_o⟩}) <s (𝑍 ↾ suc dom 𝑈))
200		nosepssdm 33816	. . . . 5 ⊢ ((𝑈 ∈ No ∧ 𝑍 ∈ No ∧ 𝑈 ≠ 𝑍) → ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ⊆ dom 𝑈)
201	7, 35, 120, 200	syl3anc 1369	. . . 4 ⊢ ((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) → ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ⊆ dom 𝑈)
202		nosepon 33795	. . . . . 6 ⊢ ((𝑈 ∈ No ∧ 𝑍 ∈ No ∧ 𝑈 ≠ 𝑍) → ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ On)
203	7, 35, 120, 202	syl3anc 1369	. . . . 5 ⊢ ((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) → ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ On)
204		onsseleq 6292	. . . . 5 ⊢ ((∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ On ∧ dom 𝑈 ∈ On) → (∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ⊆ dom 𝑈 ↔ (∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈 ∨ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈)))
205	203, 9, 204	syl2anc 583	. . . 4 ⊢ ((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) → (∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ⊆ dom 𝑈 ↔ (∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈 ∨ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈)))
206	201, 205	mpbid 231	. . 3 ⊢ ((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) → (∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈 ∨ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈))
207	99, 199, 206	mpjaodan 955	. 2 ⊢ ((((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) ∧ 𝑍 <s 𝑈) → ¬ (𝑈 ∪ {⟨dom 𝑈, 1_o⟩}) <s (𝑍 ↾ suc dom 𝑈))
208	4, 207	mpdan 683	1 ⊢ (((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) → ¬ (𝑈 ∪ {⟨dom 𝑈, 1_o⟩}) <s (𝑍 ↾ suc dom 𝑈))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 843 ∨ w3o 1084 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∀wral 3063 ∃wrex 3064 {crab 3067 ∪ cun 3881 ∩ cin 3882 ⊆ wss 3883 ∅c0 4253 {csn 4558 {ctp 4562 ⟨cop 4564 ∩ cint 4876 class class class wbr 5070 Or wor 5493 dom cdm 5580 ↾ cres 5582 Rel wrel 5585 Ord word 6250 Oncon0 6251 suc csuc 6253 Fun wfun 6412 Fn wfn 6413 ‘cfv 6418 1_oc1o 8260 2_oc2o 8261 No csur 33770 <s cslt 33771

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pr 5347 ax-un 7566

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-ord 6254 df-on 6255 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-1o 8267 df-2o 8268 df-no 33773 df-slt 33774

This theorem is referenced by: noinfbnd2 33861

W3C validator