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Theorem noinfbnd2lem1 27702

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 5103	. . 3 ⊢ (𝑏 = 𝑈 → (𝑍 <s 𝑏 ↔ 𝑍 <s 𝑈))
2		simp3 1139	. . 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 1199	. . 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 3578	. 2 ⊢ (((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) → 𝑍 <s 𝑈)
5		simpl21 1253	. . . . . . . . . . 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 1226	. . . . . . . . . . 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 3935	. . . . . . . . . 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 27622	. . . . . . . . . 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 6343	. . . . . . . . 9 ⊢ ((dom 𝑈 ∈ On ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ On)
11	9, 10	sylan 581	. . . . . . . 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 769	. . . . . . . . . . . 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 6365	. . . . . . . . . . . . 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 700	. . . . . . . . . . 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 6855	. . . . . . . . . 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 6343	. . . . . . . . . . . . 13 ⊢ ((∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ On ∧ 𝑞 ∈ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}) → 𝑞 ∈ On)
21	11, 20	sylan 581	. . . . . . . . . . . 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 6835	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑞 → (𝑈‘𝑥) = (𝑈‘𝑞))
23		fveq2 6835	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑞 → (𝑍‘𝑥) = (𝑍‘𝑞))
24	22, 23	neeq12d 2994	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑞 → ((𝑈‘𝑥) ≠ (𝑍‘𝑥) ↔ (𝑈‘𝑞) ≠ (𝑍‘𝑞)))
25	24	onnminsb 7746	. . . . . . . . . . . 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 2934	. . . . . . . . . . . 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 234	. . . . . . . . . 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 2775	. . . . . . . . 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 3129	. . . . . . . 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 6855	. . . . . . . . 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 769	. . . . . . . . . . 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 1255	. . . . . . . . . . . 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 27628	. . . . . . . . . . . 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 686	. . . . . . . . . . 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 232	. . . . . . . . . 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 2986	. . . . . . . . . . . . 13 ⊢ ((𝑈‘𝑥) ≠ (𝑍‘𝑥) ↔ (𝑍‘𝑥) ≠ (𝑈‘𝑥))
41	40	rabbii 3405	. . . . . . . . . . . 12 ⊢ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = {𝑥 ∈ On ∣ (𝑍‘𝑥) ≠ (𝑈‘𝑥)}
42	41	inteqi 4907	. . . . . . . . . . 11 ⊢ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = ∩ {𝑥 ∈ On ∣ (𝑍‘𝑥) ≠ (𝑈‘𝑥)}
43	42	fveq2i 6838	. . . . . . . . . 10 ⊢ (𝑍‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}) = (𝑍‘∩ {𝑥 ∈ On ∣ (𝑍‘𝑥) ≠ (𝑈‘𝑥)})
44	42	fveq2i 6838	. . . . . . . . . 10 ⊢ (𝑈‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}) = (𝑈‘∩ {𝑥 ∈ On ∣ (𝑍‘𝑥) ≠ (𝑈‘𝑥)})
45	39, 43, 44	3brtr4g 5133	. . . . . . . . 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 5121	. . . . . . . 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 3294	. . . . . . . . . 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 6835	. . . . . . . . . . 11 ⊢ (𝑝 = ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} → ((𝑍 ↾ dom 𝑈)‘𝑝) = ((𝑍 ↾ dom 𝑈)‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}))
49		fveq2 6835	. . . . . . . . . . 11 ⊢ (𝑝 = ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} → (𝑈‘𝑝) = (𝑈‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}))
50	48, 49	breq12d 5112	. . . . . . . . . 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 633	. . . . . . . . 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 3577	. . . . . . . 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 837	. . . . . . 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 27632	. . . . . . . . 9 ⊢ ((𝑍 ∈ No ∧ dom 𝑈 ∈ On) → (𝑍 ↾ dom 𝑈) ∈ No )
55	35, 9, 54	syl2anc 585	. . . . . . . 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 27619	. . . . . . . 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 686	. . . . . . 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 6400	. . . . . . 7 ⊢ dom 𝑈 ⊆ suc dom 𝑈
60		resabs1 5966	. . . . . . 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 5954	. . . . . . 7 ⊢ ((𝑈 ∪ {⟨dom 𝑈, 1_o⟩}) ↾ dom 𝑈) = ((𝑈 ↾ dom 𝑈) ∪ ({⟨dom 𝑈, 1_o⟩} ↾ dom 𝑈))
63		nofun 27621	. . . . . . . . . . . . 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 6510	. . . . . . . . . . . 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 5986	. . . . . . . . . 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 27625	. . . . . . . . . . . . 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 6336	. . . . . . . . . . 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 8409	. . . . . . . . . . 11 ⊢ 1_o ∈ V
76	75	snres0 6257	. . . . . . . . . 10 ⊢ (({⟨dom 𝑈, 1_o⟩} ↾ dom 𝑈) = ∅ ↔ ¬ dom 𝑈 ∈ dom 𝑈)
77	74, 76	sylibr 234	. . . . . . . . 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 4122	. . . . . . . 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 4347	. . . . . . . 8 ⊢ (𝑈 ∪ ∅) = 𝑈
80	78, 79	eqtrdi 2788	. . . . . . 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 2784	. . . . . 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 5131	. . . . 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 7761	. . . . . . . . 9 ⊢ (dom 𝑈 ∈ On ↔ suc dom 𝑈 ∈ On)
84	9, 83	sylib 218	. . . . . . . 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 27632	. . . . . . . 8 ⊢ ((𝑍 ∈ No ∧ suc dom 𝑈 ∈ On) → (𝑍 ↾ suc dom 𝑈) ∈ No )
86	35, 84, 85	syl2anc 585	. . . . . . 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 4720	. . . . . . . . 9 ⊢ 1_o ∈ {1_o, 2_o}
89	88	noextend 27638	. . . . . . . 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 27634	. . . . . 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 1374	. . . . 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 27648	. . . . . 6 ⊢ <s Or No
96		soasym 5566	. . . . . 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 691	. . . . 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 686	. . . 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 5557	. . . . . 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 588	. . . . 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 6324	. . . . . . . 8 ⊢ suc dom 𝑈 = (dom 𝑈 ∪ {dom 𝑈})
104	103	reseq2i 5936	. . . . . . 7 ⊢ (𝑍 ↾ suc dom 𝑈) = (𝑍 ↾ (dom 𝑈 ∪ {dom 𝑈}))
105		resundi 5953	. . . . . . 7 ⊢ (𝑍 ↾ (dom 𝑈 ∪ {dom 𝑈})) = ((𝑍 ↾ dom 𝑈) ∪ (𝑍 ↾ {dom 𝑈}))
106	104, 105	eqtri 2760	. . . . . 6 ⊢ (𝑍 ↾ suc dom 𝑈) = ((𝑍 ↾ dom 𝑈) ∪ (𝑍 ↾ {dom 𝑈}))
107		dmres 5972	. . . . . . . . 9 ⊢ dom (𝑍 ↾ dom 𝑈) = (dom 𝑈 ∩ dom 𝑍)
108	42	eqeq1i 2742	. . . . . . . . . . . . 13 ⊢ (∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈 ↔ ∩ {𝑥 ∈ On ∣ (𝑍‘𝑥) ≠ (𝑈‘𝑥)} = dom 𝑈)
109	108	biimpi 216	. . . . . . . . . . . 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 1210	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) → 𝑍 ∈ No )
114		sonr 5557	. . . . . . . . . . . . . . . . . 18 ⊢ (( <s Or No ∧ 𝑍 ∈ No ) → ¬ 𝑍 <s 𝑍)
115	95, 113, 114	sylancr 588	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) → ¬ 𝑍 <s 𝑍)
116		breq2 5103	. . . . . . . . . . . . . . . . . 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 247	. . . . . . . . . . . . . . . 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 2948	. . . . . . . . . . . . . . 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 2988	. . . . . . . . . . . . 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 27658	. . . . . . . . . . . 12 ⊢ ((𝑍 ∈ No ∧ 𝑈 ∈ No ∧ 𝑍 ≠ 𝑈) → ∩ {𝑥 ∈ On ∣ (𝑍‘𝑥) ≠ (𝑈‘𝑥)} ⊆ dom 𝑍)
124	111, 112, 122, 123	syl3anc 1374	. . . . . . . . . . 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 3970	. . . . . . . . . 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		dfss2 3920	. . . . . . . . . 10 ⊢ (dom 𝑈 ⊆ dom 𝑍 ↔ (dom 𝑈 ∩ dom 𝑍) = dom 𝑈)
127	125, 126	sylib 218	. . . . . . . . 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 2784	. . . . . . . 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 2823	. . . . . . . . . 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 6855	. . . . . . . . . . . 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 6343	. . . . . . . . . . . . . . 15 ⊢ ((dom 𝑈 ∈ On ∧ 𝑞 ∈ dom 𝑈) → 𝑞 ∈ On)
134	132, 133	sylan 581	. . . . . . . . . . . . . 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 2823	. . . . . . . . . . . . . . 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 2989	. . . . . . . . . . . . . 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 234	. . . . . . . . . . . 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 2772	. . . . . . . . . . 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 240	. . . . . . . . 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 3128	. . . . . . . 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 27621	. . . . . . . . . . 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 6536	. . . . . . . . 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 6983	. . . . . . . . 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 585	. . . . . . . 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 714	. . . . . . 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 6524	. . . . . . . . 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		ndmfv 6867	. . . . . . . . . . . . . . . 16 ⊢ (¬ dom 𝑈 ∈ dom 𝑈 → (𝑈‘dom 𝑈) = ∅)
156		2on0 8413	. . . . . . . . . . . . . . . . . . 19 ⊢ 2_o ≠ ∅
157	156	necomi 2987	. . . . . . . . . . . . . . . . . 18 ⊢ ∅ ≠ 2_o
158		neeq1 2995	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑈‘dom 𝑈) = ∅ → ((𝑈‘dom 𝑈) ≠ 2_o ↔ ∅ ≠ 2_o))
159	157, 158	mpbiri 258	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑈‘dom 𝑈) = ∅ → (𝑈‘dom 𝑈) ≠ 2_o)
160	159	neneqd 2938	. . . . . . . . . . . . . . . 16 ⊢ ((𝑈‘dom 𝑈) = ∅ → ¬ (𝑈‘dom 𝑈) = 2_o)
161	155, 160	syl 17	. . . . . . . . . . . . . . 15 ⊢ (¬ dom 𝑈 ∈ dom 𝑈 → ¬ (𝑈‘dom 𝑈) = 2_o)
162	112, 70, 73, 161	4syl 19	. . . . . . . . . . . . . 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)
163	162	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))
164		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)
165	35, 7, 37	syl2anc 585	. . . . . . . . . . . . . . . . 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)})))
166	164, 165	mpbid 232	. . . . . . . . . . . . . . . 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)}))
167	166	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)}))
168	110	fveq2d 6839	. . . . . . . . . . . . . . 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))
169	110	fveq2d 6839	. . . . . . . . . . . . . . 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))
170	167, 168, 169	3brtr3d 5130	. . . . . . . . . . . . . 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))
171		fvex 6848	. . . . . . . . . . . . . . 15 ⊢ (𝑍‘dom 𝑈) ∈ V
172		fvex 6848	. . . . . . . . . . . . . . 15 ⊢ (𝑈‘dom 𝑈) ∈ V
173	171, 172	brtp 5472	. . . . . . . . . . . . . 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)))
174	170, 173	sylib 218	. . . . . . . . . . . . 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)))
175		3orel3 1489	. . . . . . . . . . . . 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))))
176	163, 174, 175	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)))
177		andi 1010	. . . . . . . . . . . 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)))
178	176, 177	sylibr 234	. . . . . . . . . . 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)))
179	178	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)
180		ndmfv 6867	. . . . . . . . . . . 12 ⊢ (¬ dom 𝑈 ∈ dom 𝑍 → (𝑍‘dom 𝑈) = ∅)
181		1n0 8417	. . . . . . . . . . . . . . 15 ⊢ 1_o ≠ ∅
182	181	necomi 2987	. . . . . . . . . . . . . 14 ⊢ ∅ ≠ 1_o
183		neeq1 2995	. . . . . . . . . . . . . 14 ⊢ ((𝑍‘dom 𝑈) = ∅ → ((𝑍‘dom 𝑈) ≠ 1_o ↔ ∅ ≠ 1_o))
184	182, 183	mpbiri 258	. . . . . . . . . . . . 13 ⊢ ((𝑍‘dom 𝑈) = ∅ → (𝑍‘dom 𝑈) ≠ 1_o)
185	184	neneqd 2938	. . . . . . . . . . . 12 ⊢ ((𝑍‘dom 𝑈) = ∅ → ¬ (𝑍‘dom 𝑈) = 1_o)
186	180, 185	syl 17	. . . . . . . . . . 11 ⊢ (¬ dom 𝑈 ∈ dom 𝑍 → ¬ (𝑍‘dom 𝑈) = 1_o)
187	186	con4i 114	. . . . . . . . . 10 ⊢ ((𝑍‘dom 𝑈) = 1_o → dom 𝑈 ∈ dom 𝑍)
188	179, 187	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)
189		fnressn 7105	. . . . . . . . 9 ⊢ ((𝑍 Fn dom 𝑍 ∧ dom 𝑈 ∈ dom 𝑍) → (𝑍 ↾ {dom 𝑈}) = {⟨dom 𝑈, (𝑍‘dom 𝑈)⟩})
190	154, 188, 189	syl2an2r 686	. . . . . . . 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)⟩})
191	179	opeq2d 4837	. . . . . . . . 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⟩)
192	191	sneqd 4593	. . . . . . . 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⟩})
193	190, 192	eqtrd 2772	. . . . . . 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⟩})
194	152, 193	uneq12d 4122	. . . . . 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⟩}))
195	106, 194	eqtrid 2784	. . . . 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⟩}))
196	195	breq2d 5111	. . . 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⟩})))
197	102, 196	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))
198		nosepssdm 27658	. . . . 5 ⊢ ((𝑈 ∈ No ∧ 𝑍 ∈ No ∧ 𝑈 ≠ 𝑍) → ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ⊆ dom 𝑈)
199	7, 35, 120, 198	syl3anc 1374	. . . 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)
200		nosepon 27637	. . . . . 6 ⊢ ((𝑈 ∈ No ∧ 𝑍 ∈ No ∧ 𝑈 ≠ 𝑍) → ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ On)
201	7, 35, 120, 200	syl3anc 1374	. . . . 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)
202		onsseleq 6359	. . . . 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)))
203	201, 9, 202	syl2anc 585	. . . 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)))
204	199, 203	mpbid 232	. . 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))
205	99, 197, 204	mpjaodan 961	. 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))
206	4, 205	mpdan 688	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 206 ∧ wa 395 ∨ wo 848 ∨ w3o 1086 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 ∃wrex 3061 {crab 3400 ∪ cun 3900 ∩ cin 3901 ⊆ wss 3902 ∅c0 4286 {csn 4581 {ctp 4585 ⟨cop 4587 ∩ cint 4903 class class class wbr 5099 Or wor 5532 dom cdm 5625 ↾ cres 5627 Rel wrel 5630 Ord word 6317 Oncon0 6318 suc csuc 6320 Fun wfun 6487 Fn wfn 6488 ‘cfv 6493 1_oc1o 8392 2_oc2o 8393 No csur 27611 <s cslt 27612

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3062 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-int 4904 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-ord 6321 df-on 6322 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-1o 8399 df-2o 8400 df-no 27614 df-slt 27615

This theorem is referenced by: noinfbnd2 27703

W3C validator