noinfbnd2lem1 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > noinfbnd2lem1		Structured version Visualization version GIF version

Theorem noinfbnd2lem1 27618

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 5145	. . 3 ⊢ (𝑏 = 𝑈 → (𝑍 <s 𝑏 ↔ 𝑍 <s 𝑈))
2		simp3 1135	. . 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 1194	. . 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 3607	. 2 ⊢ (((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) → 𝑍 <s 𝑈)
5		simpl21 1248	. . . . . . . . . . 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 1221	. . . . . . . . . . 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 3978	. . . . . . . . . 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 27538	. . . . . . . . . 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 6383	. . . . . . . . 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 766	. . . . . . . . . . . 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 6404	. . . . . . . . . . . . 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 696	. . . . . . . . . . 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 6905	. . . . . . . . . 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 6383	. . . . . . . . . . . . 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 6885	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑞 → (𝑈‘𝑥) = (𝑈‘𝑞))
23		fveq2 6885	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑞 → (𝑍‘𝑥) = (𝑍‘𝑞))
24	22, 23	neeq12d 2996	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑞 → ((𝑈‘𝑥) ≠ (𝑍‘𝑥) ↔ (𝑈‘𝑞) ≠ (𝑍‘𝑞)))
25	24	onnminsb 7784	. . . . . . . . . . . 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 2935	. . . . . . . . . . . 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 2769	. . . . . . . . 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 3140	. . . . . . . 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 6905	. . . . . . . . 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 766	. . . . . . . . . . 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 1250	. . . . . . . . . . . 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 27544	. . . . . . . . . . . 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 682	. . . . . . . . . . 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 2988	. . . . . . . . . . . . 13 ⊢ ((𝑈‘𝑥) ≠ (𝑍‘𝑥) ↔ (𝑍‘𝑥) ≠ (𝑈‘𝑥))
41	40	rabbii 3432	. . . . . . . . . . . 12 ⊢ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = {𝑥 ∈ On ∣ (𝑍‘𝑥) ≠ (𝑈‘𝑥)}
42	41	inteqi 4947	. . . . . . . . . . 11 ⊢ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = ∩ {𝑥 ∈ On ∣ (𝑍‘𝑥) ≠ (𝑈‘𝑥)}
43	42	fveq2i 6888	. . . . . . . . . 10 ⊢ (𝑍‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}) = (𝑍‘∩ {𝑥 ∈ On ∣ (𝑍‘𝑥) ≠ (𝑈‘𝑥)})
44	42	fveq2i 6888	. . . . . . . . . 10 ⊢ (𝑈‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}) = (𝑈‘∩ {𝑥 ∈ On ∣ (𝑍‘𝑥) ≠ (𝑈‘𝑥)})
45	39, 43, 44	3brtr4g 5175	. . . . . . . . 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 5163	. . . . . . . 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 3316	. . . . . . . . . 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 6885	. . . . . . . . . . 11 ⊢ (𝑝 = ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} → ((𝑍 ↾ dom 𝑈)‘𝑝) = ((𝑍 ↾ dom 𝑈)‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}))
49		fveq2 6885	. . . . . . . . . . 11 ⊢ (𝑝 = ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} → (𝑈‘𝑝) = (𝑈‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}))
50	48, 49	breq12d 5154	. . . . . . . . . 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 3606	. . . . . . . 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 834	. . . . . . 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 27548	. . . . . . . . 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 27535	. . . . . . . 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 682	. . . . . . 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 257	. . . . . 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 6438	. . . . . . 7 ⊢ dom 𝑈 ⊆ suc dom 𝑈
60		resabs1 6005	. . . . . . 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 5990	. . . . . . 7 ⊢ ((𝑈 ∪ {⟨dom 𝑈, 1_o⟩}) ↾ dom 𝑈) = ((𝑈 ↾ dom 𝑈) ∪ ({⟨dom 𝑈, 1_o⟩} ↾ dom 𝑈))
63		nofun 27537	. . . . . . . . . . . . 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 6559	. . . . . . . . . . . 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 6020	. . . . . . . . . 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 27541	. . . . . . . . . . . . 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 6376	. . . . . . . . . . 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 8477	. . . . . . . . . . 11 ⊢ 1_o ∈ V
76	75	snres0 6291	. . . . . . . . . 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 4159	. . . . . . . 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 4385	. . . . . . . 8 ⊢ (𝑈 ∪ ∅) = 𝑈
80	78, 79	eqtrdi 2782	. . . . . . 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	eqtrid 2778	. . . . . 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 5173	. . . . 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		onsucb 7802	. . . . . . . . 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 27548	. . . . . . . 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 4761	. . . . . . . . 9 ⊢ 1_o ∈ {1_o, 2_o}
89	88	noextend 27554	. . . . . . . 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 27550	. . . . . 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 1368	. . . . 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 27564	. . . . . 6 ⊢ <s Or No
96		soasym 5612	. . . . . 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 687	. . . . 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 682	. . . 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 5604	. . . . . 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 6364	. . . . . . . 8 ⊢ suc dom 𝑈 = (dom 𝑈 ∪ {dom 𝑈})
104	103	reseq2i 5972	. . . . . . 7 ⊢ (𝑍 ↾ suc dom 𝑈) = (𝑍 ↾ (dom 𝑈 ∪ {dom 𝑈}))
105		resundi 5989	. . . . . . 7 ⊢ (𝑍 ↾ (dom 𝑈 ∪ {dom 𝑈})) = ((𝑍 ↾ dom 𝑈) ∪ (𝑍 ↾ {dom 𝑈}))
106	104, 105	eqtri 2754	. . . . . 6 ⊢ (𝑍 ↾ suc dom 𝑈) = ((𝑍 ↾ dom 𝑈) ∪ (𝑍 ↾ {dom 𝑈}))
107		dmres 5997	. . . . . . . . 9 ⊢ dom (𝑍 ↾ dom 𝑈) = (dom 𝑈 ∩ dom 𝑍)
108	42	eqeq1i 2731	. . . . . . . . . . . . 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 1205	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) → 𝑍 ∈ No )
114		sonr 5604	. . . . . . . . . . . . . . . . . 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 5145	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑈 = 𝑍 → (𝑍 <s 𝑈 ↔ 𝑍 <s 𝑍))
117	116	notbid 318	. . . . . . . . . . . . . . . . 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 2949	. . . . . . . . . . . . . . 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 2990	. . . . . . . . . . . . 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 27574	. . . . . . . . . . . 12 ⊢ ((𝑍 ∈ No ∧ 𝑈 ∈ No ∧ 𝑍 ≠ 𝑈) → ∩ {𝑥 ∈ On ∣ (𝑍‘𝑥) ≠ (𝑈‘𝑥)} ⊆ dom 𝑍)
124	111, 112, 122, 123	syl3anc 1368	. . . . . . . . . . 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 4016	. . . . . . . . . 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 3960	. . . . . . . . . 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	eqtrid 2778	. . . . . . . 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 2813	. . . . . . . . . 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 6905	. . . . . . . . . . . 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 6383	. . . . . . . . . . . . . . 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 2813	. . . . . . . . . . . . . . 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 2991	. . . . . . . . . . . . . 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 2766	. . . . . . . . . . 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 3139	. . . . . . . 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 27537	. . . . . . . . . . 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 6585	. . . . . . . . 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 7031	. . . . . . . . 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 710	. . . . . . 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 6573	. . . . . . . . 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 6920	. . . . . . . . . . . . . . . 16 ⊢ (¬ dom 𝑈 ∈ dom 𝑈 → (𝑈‘dom 𝑈) = ∅)
158		2on0 8483	. . . . . . . . . . . . . . . . . . 19 ⊢ 2_o ≠ ∅
159	158	necomi 2989	. . . . . . . . . . . . . . . . . 18 ⊢ ∅ ≠ 2_o
160		neeq1 2997	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑈‘dom 𝑈) = ∅ → ((𝑈‘dom 𝑈) ≠ 2_o ↔ ∅ ≠ 2_o))
161	159, 160	mpbiri 258	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑈‘dom 𝑈) = ∅ → (𝑈‘dom 𝑈) ≠ 2_o)
162	161	neneqd 2939	. . . . . . . . . . . . . . . 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 6889	. . . . . . . . . . . . . . 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 6889	. . . . . . . . . . . . . . 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 5172	. . . . . . . . . . . . . 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 6898	. . . . . . . . . . . . . . 15 ⊢ (𝑍‘dom 𝑈) ∈ V
174		fvex 6898	. . . . . . . . . . . . . . 15 ⊢ (𝑈‘dom 𝑈) ∈ V
175	173, 174	brtp 5516	. . . . . . . . . . . . . 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 1481	. . . . . . . . . . . . 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 6920	. . . . . . . . . . . 12 ⊢ (¬ dom 𝑈 ∈ dom 𝑍 → (𝑍‘dom 𝑈) = ∅)
183		1n0 8489	. . . . . . . . . . . . . . 15 ⊢ 1_o ≠ ∅
184	183	necomi 2989	. . . . . . . . . . . . . 14 ⊢ ∅ ≠ 1_o
185		neeq1 2997	. . . . . . . . . . . . . 14 ⊢ ((𝑍‘dom 𝑈) = ∅ → ((𝑍‘dom 𝑈) ≠ 1_o ↔ ∅ ≠ 1_o))
186	184, 185	mpbiri 258	. . . . . . . . . . . . 13 ⊢ ((𝑍‘dom 𝑈) = ∅ → (𝑍‘dom 𝑈) ≠ 1_o)
187	186	neneqd 2939	. . . . . . . . . . . 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 7152	. . . . . . . . 9 ⊢ ((𝑍 Fn dom 𝑍 ∧ dom 𝑈 ∈ dom 𝑍) → (𝑍 ↾ {dom 𝑈}) = {⟨dom 𝑈, (𝑍‘dom 𝑈)⟩})
192	154, 190, 191	syl2an2r 682	. . . . . . . 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 4875	. . . . . . . . 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 4635	. . . . . . . 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 2766	. . . . . . 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 4159	. . . . . 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	eqtrid 2778	. . . . 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 5153	. . . 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 325	. . 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 27574	. . . . 5 ⊢ ((𝑈 ∈ No ∧ 𝑍 ∈ No ∧ 𝑈 ≠ 𝑍) → ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ⊆ dom 𝑈)
201	7, 35, 120, 200	syl3anc 1368	. . . 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 27553	. . . . . 6 ⊢ ((𝑈 ∈ No ∧ 𝑍 ∈ No ∧ 𝑈 ≠ 𝑍) → ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ On)
203	7, 35, 120, 202	syl3anc 1368	. . . . 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 6399	. . . . 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 684	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 844 ∨ w3o 1083 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 ∀wral 3055 ∃wrex 3064 {crab 3426 ∪ cun 3941 ∩ cin 3942 ⊆ wss 3943 ∅c0 4317 {csn 4623 {ctp 4627 ⟨cop 4629 ∩ cint 4943 class class class wbr 5141 Or wor 5580 dom cdm 5669 ↾ cres 5671 Rel wrel 5674 Ord word 6357 Oncon0 6358 suc csuc 6360 Fun wfun 6531 Fn wfn 6532 ‘cfv 6537 1_oc1o 8460 2_oc2o 8461 No csur 27528 <s cslt 27529

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7722

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-ord 6361 df-on 6362 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-1o 8467 df-2o 8468 df-no 27531 df-slt 27532

This theorem is referenced by: noinfbnd2 27619

W3C validator