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Theorem nosupbnd2lem1 27564

Description: Bounding law from above when a set of surreals has a maximum. (Contributed by Scott Fenton, 6-Dec-2021.)

**Assertion**
Ref	Expression
nosupbnd2lem1	⊢ (((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) → ¬ (𝑍 ↾ suc dom 𝑈) <s (𝑈 ∪ {⟨dom 𝑈, 2_o⟩}))

Distinct variable groups: 𝐴,𝑎 𝑈,𝑎 𝑍,𝑎 Allowed substitution hints: 𝐴(𝑦) 𝑈(𝑦) 𝑍(𝑦)

Proof of Theorem nosupbnd2lem1 Dummy variables 𝑞 𝑝 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1l 1194	. . 3 ⊢ (((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) → 𝑈 ∈ 𝐴)
2		simp3 1135	. . 3 ⊢ (((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) → ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍)
3		breq1 5141	. . . 4 ⊢ (𝑎 = 𝑈 → (𝑎 <s 𝑍 ↔ 𝑈 <s 𝑍))
4	3	rspcv 3600	. . 3 ⊢ (𝑈 ∈ 𝐴 → (∀𝑎 ∈ 𝐴 𝑎 <s 𝑍 → 𝑈 <s 𝑍))
5	1, 2, 4	sylc 65	. 2 ⊢ (((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) → 𝑈 <s 𝑍)
6		simpl21 1248	. . . . 5 ⊢ ((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) → 𝐴 ⊆ No )
7		simpl1l 1221	. . . . 5 ⊢ ((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) → 𝑈 ∈ 𝐴)
8	6, 7	sseldd 3975	. . . 4 ⊢ ((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) → 𝑈 ∈ No )
9		simpl23 1250	. . . 4 ⊢ ((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) → 𝑍 ∈ No )
10		simp21 1203	. . . . . . . . . 10 ⊢ (((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) → 𝐴 ⊆ No )
11	10, 1	sseldd 3975	. . . . . . . . 9 ⊢ (((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) → 𝑈 ∈ No )
12		sltso 27525	. . . . . . . . . 10 ⊢ <s Or No
13		sonr 5601	. . . . . . . . . 10 ⊢ (( <s Or No ∧ 𝑈 ∈ No ) → ¬ 𝑈 <s 𝑈)
14	12, 13	mpan 687	. . . . . . . . 9 ⊢ (𝑈 ∈ No → ¬ 𝑈 <s 𝑈)
15	11, 14	syl 17	. . . . . . . 8 ⊢ (((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) → ¬ 𝑈 <s 𝑈)
16		breq2 5142	. . . . . . . . 9 ⊢ (𝑈 = 𝑍 → (𝑈 <s 𝑈 ↔ 𝑈 <s 𝑍))
17	16	notbid 318	. . . . . . . 8 ⊢ (𝑈 = 𝑍 → (¬ 𝑈 <s 𝑈 ↔ ¬ 𝑈 <s 𝑍))
18	15, 17	syl5ibcom 244	. . . . . . 7 ⊢ (((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) → (𝑈 = 𝑍 → ¬ 𝑈 <s 𝑍))
19	18	con2d 134	. . . . . 6 ⊢ (((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) → (𝑈 <s 𝑍 → ¬ 𝑈 = 𝑍))
20	19	imp 406	. . . . 5 ⊢ ((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) → ¬ 𝑈 = 𝑍)
21	20	neqned 2939	. . . 4 ⊢ ((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) → 𝑈 ≠ 𝑍)
22		nosepssdm 27535	. . . 4 ⊢ ((𝑈 ∈ No ∧ 𝑍 ∈ No ∧ 𝑈 ≠ 𝑍) → ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ⊆ dom 𝑈)
23	8, 9, 21, 22	syl3anc 1368	. . 3 ⊢ ((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) → ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ⊆ dom 𝑈)
24		nosepon 27514	. . . . . 6 ⊢ ((𝑈 ∈ No ∧ 𝑍 ∈ No ∧ 𝑈 ≠ 𝑍) → ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ On)
25	8, 9, 21, 24	syl3anc 1368	. . . . 5 ⊢ ((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) → ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ On)
26		nodmon 27499	. . . . . 6 ⊢ (𝑈 ∈ No → dom 𝑈 ∈ On)
27	8, 26	syl 17	. . . . 5 ⊢ ((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) → dom 𝑈 ∈ On)
28		onsseleq 6395	. . . . 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)))
29	25, 27, 28	syl2anc 583	. . . 4 ⊢ ((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) → (∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ⊆ dom 𝑈 ↔ (∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈 ∨ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈)))
30	8	adantr 480	. . . . . . . . . . . 12 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → 𝑈 ∈ No )
31	9	adantr 480	. . . . . . . . . . . 12 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → 𝑍 ∈ No )
32	21	adantr 480	. . . . . . . . . . . 12 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → 𝑈 ≠ 𝑍)
33	30, 31, 32, 24	syl3anc 1368	. . . . . . . . . . 11 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ On)
34		onelon 6379	. . . . . . . . . . . . . . . 16 ⊢ ((∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ On ∧ 𝑞 ∈ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}) → 𝑞 ∈ On)
35	33, 34	sylan 579	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) ∧ 𝑞 ∈ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}) → 𝑞 ∈ On)
36		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) ∧ 𝑞 ∈ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}) → 𝑞 ∈ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)})
37		fveq2 6881	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑞 → (𝑈‘𝑥) = (𝑈‘𝑞))
38		fveq2 6881	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑞 → (𝑍‘𝑥) = (𝑍‘𝑞))
39	37, 38	neeq12d 2994	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑞 → ((𝑈‘𝑥) ≠ (𝑍‘𝑥) ↔ (𝑈‘𝑞) ≠ (𝑍‘𝑞)))
40	39	onnminsb 7780	. . . . . . . . . . . . . . 15 ⊢ (𝑞 ∈ On → (𝑞 ∈ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} → ¬ (𝑈‘𝑞) ≠ (𝑍‘𝑞)))
41	35, 36, 40	sylc 65	. . . . . . . . . . . . . 14 ⊢ ((((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) ∧ 𝑞 ∈ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}) → ¬ (𝑈‘𝑞) ≠ (𝑍‘𝑞))
42		df-ne 2933	. . . . . . . . . . . . . . 15 ⊢ ((𝑈‘𝑞) ≠ (𝑍‘𝑞) ↔ ¬ (𝑈‘𝑞) = (𝑍‘𝑞))
43	42	con2bii 357	. . . . . . . . . . . . . 14 ⊢ ((𝑈‘𝑞) = (𝑍‘𝑞) ↔ ¬ (𝑈‘𝑞) ≠ (𝑍‘𝑞))
44	41, 43	sylibr 233	. . . . . . . . . . . . 13 ⊢ ((((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) ∧ 𝑞 ∈ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}) → (𝑈‘𝑞) = (𝑍‘𝑞))
45		simplr 766	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) ∧ 𝑞 ∈ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}) → ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈)
46	27	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → dom 𝑈 ∈ On)
47	46	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) ∧ 𝑞 ∈ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}) → dom 𝑈 ∈ On)
48		ontr1 6400	. . . . . . . . . . . . . . . 16 ⊢ (dom 𝑈 ∈ On → ((𝑞 ∈ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → 𝑞 ∈ dom 𝑈))
49	47, 48	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {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))
50	36, 45, 49	mp2and 696	. . . . . . . . . . . . . 14 ⊢ ((((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) ∧ 𝑞 ∈ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}) → 𝑞 ∈ dom 𝑈)
51	50	fvresd 6901	. . . . . . . . . . . . 13 ⊢ ((((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) ∧ 𝑞 ∈ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}) → ((𝑍 ↾ dom 𝑈)‘𝑞) = (𝑍‘𝑞))
52	44, 51	eqtr4d 2767	. . . . . . . . . . . 12 ⊢ ((((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) ∧ 𝑞 ∈ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}) → (𝑈‘𝑞) = ((𝑍 ↾ dom 𝑈)‘𝑞))
53	52	ralrimiva 3138	. . . . . . . . . . 11 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → ∀𝑞 ∈ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} (𝑈‘𝑞) = ((𝑍 ↾ dom 𝑈)‘𝑞))
54		simplr 766	. . . . . . . . . . . . 13 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → 𝑈 <s 𝑍)
55		sltval2 27505	. . . . . . . . . . . . . 14 ⊢ ((𝑈 ∈ No ∧ 𝑍 ∈ No ) → (𝑈 <s 𝑍 ↔ (𝑈‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}){⟨1_o, ∅⟩, ⟨1_o, 2_o⟩, ⟨∅, 2_o⟩} (𝑍‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)})))
56	30, 31, 55	syl2anc 583	. . . . . . . . . . . . 13 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → (𝑈 <s 𝑍 ↔ (𝑈‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}){⟨1_o, ∅⟩, ⟨1_o, 2_o⟩, ⟨∅, 2_o⟩} (𝑍‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)})))
57	54, 56	mpbid 231	. . . . . . . . . . . 12 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → (𝑈‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}){⟨1_o, ∅⟩, ⟨1_o, 2_o⟩, ⟨∅, 2_o⟩} (𝑍‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}))
58		simpr 484	. . . . . . . . . . . . 13 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈)
59	58	fvresd 6901	. . . . . . . . . . . 12 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → ((𝑍 ↾ dom 𝑈)‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}) = (𝑍‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}))
60	57, 59	breqtrrd 5166	. . . . . . . . . . 11 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → (𝑈‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}){⟨1_o, ∅⟩, ⟨1_o, 2_o⟩, ⟨∅, 2_o⟩} ((𝑍 ↾ dom 𝑈)‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}))
61		raleq 3314	. . . . . . . . . . . . 13 ⊢ (𝑝 = ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} → (∀𝑞 ∈ 𝑝 (𝑈‘𝑞) = ((𝑍 ↾ dom 𝑈)‘𝑞) ↔ ∀𝑞 ∈ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} (𝑈‘𝑞) = ((𝑍 ↾ dom 𝑈)‘𝑞)))
62		fveq2 6881	. . . . . . . . . . . . . 14 ⊢ (𝑝 = ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} → (𝑈‘𝑝) = (𝑈‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}))
63		fveq2 6881	. . . . . . . . . . . . . 14 ⊢ (𝑝 = ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} → ((𝑍 ↾ dom 𝑈)‘𝑝) = ((𝑍 ↾ dom 𝑈)‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}))
64	62, 63	breq12d 5151	. . . . . . . . . . . . 13 ⊢ (𝑝 = ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} → ((𝑈‘𝑝){⟨1_o, ∅⟩, ⟨1_o, 2_o⟩, ⟨∅, 2_o⟩} ((𝑍 ↾ dom 𝑈)‘𝑝) ↔ (𝑈‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}){⟨1_o, ∅⟩, ⟨1_o, 2_o⟩, ⟨∅, 2_o⟩} ((𝑍 ↾ dom 𝑈)‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)})))
65	61, 64	anbi12d 630	. . . . . . . . . . . 12 ⊢ (𝑝 = ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} → ((∀𝑞 ∈ 𝑝 (𝑈‘𝑞) = ((𝑍 ↾ dom 𝑈)‘𝑞) ∧ (𝑈‘𝑝){⟨1_o, ∅⟩, ⟨1_o, 2_o⟩, ⟨∅, 2_o⟩} ((𝑍 ↾ dom 𝑈)‘𝑝)) ↔ (∀𝑞 ∈ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} (𝑈‘𝑞) = ((𝑍 ↾ dom 𝑈)‘𝑞) ∧ (𝑈‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}){⟨1_o, ∅⟩, ⟨1_o, 2_o⟩, ⟨∅, 2_o⟩} ((𝑍 ↾ dom 𝑈)‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}))))
66	65	rspcev 3604	. . . . . . . . . . 11 ⊢ ((∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ On ∧ (∀𝑞 ∈ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} (𝑈‘𝑞) = ((𝑍 ↾ dom 𝑈)‘𝑞) ∧ (𝑈‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}){⟨1_o, ∅⟩, ⟨1_o, 2_o⟩, ⟨∅, 2_o⟩} ((𝑍 ↾ dom 𝑈)‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}))) → ∃𝑝 ∈ On (∀𝑞 ∈ 𝑝 (𝑈‘𝑞) = ((𝑍 ↾ dom 𝑈)‘𝑞) ∧ (𝑈‘𝑝){⟨1_o, ∅⟩, ⟨1_o, 2_o⟩, ⟨∅, 2_o⟩} ((𝑍 ↾ dom 𝑈)‘𝑝)))
67	33, 53, 60, 66	syl12anc 834	. . . . . . . . . 10 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → ∃𝑝 ∈ On (∀𝑞 ∈ 𝑝 (𝑈‘𝑞) = ((𝑍 ↾ dom 𝑈)‘𝑞) ∧ (𝑈‘𝑝){⟨1_o, ∅⟩, ⟨1_o, 2_o⟩, ⟨∅, 2_o⟩} ((𝑍 ↾ dom 𝑈)‘𝑝)))
68		noreson 27509	. . . . . . . . . . . 12 ⊢ ((𝑍 ∈ No ∧ dom 𝑈 ∈ On) → (𝑍 ↾ dom 𝑈) ∈ No )
69	31, 46, 68	syl2anc 583	. . . . . . . . . . 11 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → (𝑍 ↾ dom 𝑈) ∈ No )
70		sltval 27496	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ No ∧ (𝑍 ↾ dom 𝑈) ∈ No ) → (𝑈 <s (𝑍 ↾ dom 𝑈) ↔ ∃𝑝 ∈ On (∀𝑞 ∈ 𝑝 (𝑈‘𝑞) = ((𝑍 ↾ dom 𝑈)‘𝑞) ∧ (𝑈‘𝑝){⟨1_o, ∅⟩, ⟨1_o, 2_o⟩, ⟨∅, 2_o⟩} ((𝑍 ↾ dom 𝑈)‘𝑝))))
71	30, 69, 70	syl2anc 583	. . . . . . . . . 10 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → (𝑈 <s (𝑍 ↾ dom 𝑈) ↔ ∃𝑝 ∈ On (∀𝑞 ∈ 𝑝 (𝑈‘𝑞) = ((𝑍 ↾ dom 𝑈)‘𝑞) ∧ (𝑈‘𝑝){⟨1_o, ∅⟩, ⟨1_o, 2_o⟩, ⟨∅, 2_o⟩} ((𝑍 ↾ dom 𝑈)‘𝑝))))
72	67, 71	mpbird 257	. . . . . . . . 9 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → 𝑈 <s (𝑍 ↾ dom 𝑈))
73		df-res 5678	. . . . . . . . . . . . 13 ⊢ ({⟨dom 𝑈, 2_o⟩} ↾ dom 𝑈) = ({⟨dom 𝑈, 2_o⟩} ∩ (dom 𝑈 × V))
74		2on 8475	. . . . . . . . . . . . . . . 16 ⊢ 2_o ∈ On
75		xpsng 7129	. . . . . . . . . . . . . . . 16 ⊢ ((dom 𝑈 ∈ On ∧ 2_o ∈ On) → ({dom 𝑈} × {2_o}) = {⟨dom 𝑈, 2_o⟩})
76	46, 74, 75	sylancl 585	. . . . . . . . . . . . . . 15 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → ({dom 𝑈} × {2_o}) = {⟨dom 𝑈, 2_o⟩})
77	76	ineq1d 4203	. . . . . . . . . . . . . 14 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → (({dom 𝑈} × {2_o}) ∩ (dom 𝑈 × V)) = ({⟨dom 𝑈, 2_o⟩} ∩ (dom 𝑈 × V)))
78		incom 4193	. . . . . . . . . . . . . . . 16 ⊢ ({dom 𝑈} ∩ dom 𝑈) = (dom 𝑈 ∩ {dom 𝑈})
79		nodmord 27502	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑈 ∈ No → Ord dom 𝑈)
80		ordirr 6372	. . . . . . . . . . . . . . . . . 18 ⊢ (Ord dom 𝑈 → ¬ dom 𝑈 ∈ dom 𝑈)
81	30, 79, 80	3syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → ¬ dom 𝑈 ∈ dom 𝑈)
82		disjsn 4707	. . . . . . . . . . . . . . . . 17 ⊢ ((dom 𝑈 ∩ {dom 𝑈}) = ∅ ↔ ¬ dom 𝑈 ∈ dom 𝑈)
83	81, 82	sylibr 233	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → (dom 𝑈 ∩ {dom 𝑈}) = ∅)
84	78, 83	eqtrid 2776	. . . . . . . . . . . . . . 15 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → ({dom 𝑈} ∩ dom 𝑈) = ∅)
85		xpdisj1 6150	. . . . . . . . . . . . . . 15 ⊢ (({dom 𝑈} ∩ dom 𝑈) = ∅ → (({dom 𝑈} × {2_o}) ∩ (dom 𝑈 × V)) = ∅)
86	84, 85	syl 17	. . . . . . . . . . . . . 14 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → (({dom 𝑈} × {2_o}) ∩ (dom 𝑈 × V)) = ∅)
87	77, 86	eqtr3d 2766	. . . . . . . . . . . . 13 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → ({⟨dom 𝑈, 2_o⟩} ∩ (dom 𝑈 × V)) = ∅)
88	73, 87	eqtrid 2776	. . . . . . . . . . . 12 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → ({⟨dom 𝑈, 2_o⟩} ↾ dom 𝑈) = ∅)
89	88	uneq2d 4155	. . . . . . . . . . 11 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → ((𝑈 ↾ dom 𝑈) ∪ ({⟨dom 𝑈, 2_o⟩} ↾ dom 𝑈)) = ((𝑈 ↾ dom 𝑈) ∪ ∅))
90		resundir 5986	. . . . . . . . . . 11 ⊢ ((𝑈 ∪ {⟨dom 𝑈, 2_o⟩}) ↾ dom 𝑈) = ((𝑈 ↾ dom 𝑈) ∪ ({⟨dom 𝑈, 2_o⟩} ↾ dom 𝑈))
91		un0 4382	. . . . . . . . . . . 12 ⊢ ((𝑈 ↾ dom 𝑈) ∪ ∅) = (𝑈 ↾ dom 𝑈)
92	91	eqcomi 2733	. . . . . . . . . . 11 ⊢ (𝑈 ↾ dom 𝑈) = ((𝑈 ↾ dom 𝑈) ∪ ∅)
93	89, 90, 92	3eqtr4g 2789	. . . . . . . . . 10 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → ((𝑈 ∪ {⟨dom 𝑈, 2_o⟩}) ↾ dom 𝑈) = (𝑈 ↾ dom 𝑈))
94		nofun 27498	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ No → Fun 𝑈)
95	30, 94	syl 17	. . . . . . . . . . 11 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → Fun 𝑈)
96		funrel 6555	. . . . . . . . . . 11 ⊢ (Fun 𝑈 → Rel 𝑈)
97		resdm 6016	. . . . . . . . . . 11 ⊢ (Rel 𝑈 → (𝑈 ↾ dom 𝑈) = 𝑈)
98	95, 96, 97	3syl 18	. . . . . . . . . 10 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → (𝑈 ↾ dom 𝑈) = 𝑈)
99	93, 98	eqtrd 2764	. . . . . . . . 9 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → ((𝑈 ∪ {⟨dom 𝑈, 2_o⟩}) ↾ dom 𝑈) = 𝑈)
100		sssucid 6434	. . . . . . . . . 10 ⊢ dom 𝑈 ⊆ suc dom 𝑈
101		resabs1 6001	. . . . . . . . . 10 ⊢ (dom 𝑈 ⊆ suc dom 𝑈 → ((𝑍 ↾ suc dom 𝑈) ↾ dom 𝑈) = (𝑍 ↾ dom 𝑈))
102	100, 101	mp1i 13	. . . . . . . . 9 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → ((𝑍 ↾ suc dom 𝑈) ↾ dom 𝑈) = (𝑍 ↾ dom 𝑈))
103	72, 99, 102	3brtr4d 5170	. . . . . . . 8 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → ((𝑈 ∪ {⟨dom 𝑈, 2_o⟩}) ↾ dom 𝑈) <s ((𝑍 ↾ suc dom 𝑈) ↾ dom 𝑈))
104	74	elexi 3486	. . . . . . . . . . . . 13 ⊢ 2_o ∈ V
105	104	prid2 4759	. . . . . . . . . . . 12 ⊢ 2_o ∈ {1_o, 2_o}
106	105	noextend 27515	. . . . . . . . . . 11 ⊢ (𝑈 ∈ No → (𝑈 ∪ {⟨dom 𝑈, 2_o⟩}) ∈ No )
107	8, 106	syl 17	. . . . . . . . . 10 ⊢ ((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) → (𝑈 ∪ {⟨dom 𝑈, 2_o⟩}) ∈ No )
108	107	adantr 480	. . . . . . . . 9 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → (𝑈 ∪ {⟨dom 𝑈, 2_o⟩}) ∈ No )
109		onsucb 7798	. . . . . . . . . . . 12 ⊢ (dom 𝑈 ∈ On ↔ suc dom 𝑈 ∈ On)
110	27, 109	sylib 217	. . . . . . . . . . 11 ⊢ ((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) → suc dom 𝑈 ∈ On)
111		noreson 27509	. . . . . . . . . . 11 ⊢ ((𝑍 ∈ No ∧ suc dom 𝑈 ∈ On) → (𝑍 ↾ suc dom 𝑈) ∈ No )
112	9, 110, 111	syl2anc 583	. . . . . . . . . 10 ⊢ ((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) → (𝑍 ↾ suc dom 𝑈) ∈ No )
113	112	adantr 480	. . . . . . . . 9 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → (𝑍 ↾ suc dom 𝑈) ∈ No )
114		sltres 27511	. . . . . . . . 9 ⊢ (((𝑈 ∪ {⟨dom 𝑈, 2_o⟩}) ∈ No ∧ (𝑍 ↾ suc dom 𝑈) ∈ No ∧ dom 𝑈 ∈ On) → (((𝑈 ∪ {⟨dom 𝑈, 2_o⟩}) ↾ dom 𝑈) <s ((𝑍 ↾ suc dom 𝑈) ↾ dom 𝑈) → (𝑈 ∪ {⟨dom 𝑈, 2_o⟩}) <s (𝑍 ↾ suc dom 𝑈)))
115	108, 113, 46, 114	syl3anc 1368	. . . . . . . 8 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → (((𝑈 ∪ {⟨dom 𝑈, 2_o⟩}) ↾ dom 𝑈) <s ((𝑍 ↾ suc dom 𝑈) ↾ dom 𝑈) → (𝑈 ∪ {⟨dom 𝑈, 2_o⟩}) <s (𝑍 ↾ suc dom 𝑈)))
116	103, 115	mpd 15	. . . . . . 7 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → (𝑈 ∪ {⟨dom 𝑈, 2_o⟩}) <s (𝑍 ↾ suc dom 𝑈))
117		soasym 5609	. . . . . . . . 9 ⊢ (( <s Or No ∧ ((𝑈 ∪ {⟨dom 𝑈, 2_o⟩}) ∈ No ∧ (𝑍 ↾ suc dom 𝑈) ∈ No )) → ((𝑈 ∪ {⟨dom 𝑈, 2_o⟩}) <s (𝑍 ↾ suc dom 𝑈) → ¬ (𝑍 ↾ suc dom 𝑈) <s (𝑈 ∪ {⟨dom 𝑈, 2_o⟩})))
118	12, 117	mpan 687	. . . . . . . 8 ⊢ (((𝑈 ∪ {⟨dom 𝑈, 2_o⟩}) ∈ No ∧ (𝑍 ↾ suc dom 𝑈) ∈ No ) → ((𝑈 ∪ {⟨dom 𝑈, 2_o⟩}) <s (𝑍 ↾ suc dom 𝑈) → ¬ (𝑍 ↾ suc dom 𝑈) <s (𝑈 ∪ {⟨dom 𝑈, 2_o⟩})))
119	108, 113, 118	syl2anc 583	. . . . . . 7 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → ((𝑈 ∪ {⟨dom 𝑈, 2_o⟩}) <s (𝑍 ↾ suc dom 𝑈) → ¬ (𝑍 ↾ suc dom 𝑈) <s (𝑈 ∪ {⟨dom 𝑈, 2_o⟩})))
120	116, 119	mpd 15	. . . . . 6 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈) → ¬ (𝑍 ↾ suc dom 𝑈) <s (𝑈 ∪ {⟨dom 𝑈, 2_o⟩}))
121		df-suc 6360	. . . . . . . . . 10 ⊢ suc dom 𝑈 = (dom 𝑈 ∪ {dom 𝑈})
122	121	reseq2i 5968	. . . . . . . . 9 ⊢ (𝑍 ↾ suc dom 𝑈) = (𝑍 ↾ (dom 𝑈 ∪ {dom 𝑈}))
123		resundi 5985	. . . . . . . . 9 ⊢ (𝑍 ↾ (dom 𝑈 ∪ {dom 𝑈})) = ((𝑍 ↾ dom 𝑈) ∪ (𝑍 ↾ {dom 𝑈}))
124	122, 123	eqtri 2752	. . . . . . . 8 ⊢ (𝑍 ↾ suc dom 𝑈) = ((𝑍 ↾ dom 𝑈) ∪ (𝑍 ↾ {dom 𝑈}))
125		dmres 5993	. . . . . . . . . . 11 ⊢ dom (𝑍 ↾ dom 𝑈) = (dom 𝑈 ∩ dom 𝑍)
126		simpr 484	. . . . . . . . . . . . 13 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈)
127		necom 2986	. . . . . . . . . . . . . . . 16 ⊢ ((𝑈‘𝑥) ≠ (𝑍‘𝑥) ↔ (𝑍‘𝑥) ≠ (𝑈‘𝑥))
128	127	rabbii 3430	. . . . . . . . . . . . . . 15 ⊢ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = {𝑥 ∈ On ∣ (𝑍‘𝑥) ≠ (𝑈‘𝑥)}
129	128	inteqi 4944	. . . . . . . . . . . . . 14 ⊢ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = ∩ {𝑥 ∈ On ∣ (𝑍‘𝑥) ≠ (𝑈‘𝑥)}
130	9	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → 𝑍 ∈ No )
131	8	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → 𝑈 ∈ No )
132	21	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → 𝑈 ≠ 𝑍)
133	132	necomd 2988	. . . . . . . . . . . . . . 15 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → 𝑍 ≠ 𝑈)
134		nosepssdm 27535	. . . . . . . . . . . . . . 15 ⊢ ((𝑍 ∈ No ∧ 𝑈 ∈ No ∧ 𝑍 ≠ 𝑈) → ∩ {𝑥 ∈ On ∣ (𝑍‘𝑥) ≠ (𝑈‘𝑥)} ⊆ dom 𝑍)
135	130, 131, 133, 134	syl3anc 1368	. . . . . . . . . . . . . 14 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → ∩ {𝑥 ∈ On ∣ (𝑍‘𝑥) ≠ (𝑈‘𝑥)} ⊆ dom 𝑍)
136	129, 135	eqsstrid 4022	. . . . . . . . . . . . 13 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ⊆ dom 𝑍)
137	126, 136	eqsstrrd 4013	. . . . . . . . . . . 12 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → dom 𝑈 ⊆ dom 𝑍)
138		df-ss 3957	. . . . . . . . . . . 12 ⊢ (dom 𝑈 ⊆ dom 𝑍 ↔ (dom 𝑈 ∩ dom 𝑍) = dom 𝑈)
139	137, 138	sylib 217	. . . . . . . . . . 11 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → (dom 𝑈 ∩ dom 𝑍) = dom 𝑈)
140	125, 139	eqtrid 2776	. . . . . . . . . 10 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → dom (𝑍 ↾ dom 𝑈) = dom 𝑈)
141	140	eleq2d 2811	. . . . . . . . . . . 12 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → (𝑞 ∈ dom (𝑍 ↾ dom 𝑈) ↔ 𝑞 ∈ dom 𝑈))
142		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) ∧ 𝑞 ∈ dom 𝑈) → 𝑞 ∈ dom 𝑈)
143	142	fvresd 6901	. . . . . . . . . . . . . 14 ⊢ ((((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) ∧ 𝑞 ∈ dom 𝑈) → ((𝑍 ↾ dom 𝑈)‘𝑞) = (𝑍‘𝑞))
144	131, 26	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → dom 𝑈 ∈ On)
145		onelon 6379	. . . . . . . . . . . . . . . . 17 ⊢ ((dom 𝑈 ∈ On ∧ 𝑞 ∈ dom 𝑈) → 𝑞 ∈ On)
146	144, 145	sylan 579	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) ∧ 𝑞 ∈ dom 𝑈) → 𝑞 ∈ On)
147	126	eleq2d 2811	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → (𝑞 ∈ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ↔ 𝑞 ∈ dom 𝑈))
148	147	biimpar 477	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) ∧ 𝑞 ∈ dom 𝑈) → 𝑞 ∈ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)})
149	146, 148, 40	sylc 65	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) ∧ 𝑞 ∈ dom 𝑈) → ¬ (𝑈‘𝑞) ≠ (𝑍‘𝑞))
150		nesym 2989	. . . . . . . . . . . . . . . 16 ⊢ ((𝑈‘𝑞) ≠ (𝑍‘𝑞) ↔ ¬ (𝑍‘𝑞) = (𝑈‘𝑞))
151	150	con2bii 357	. . . . . . . . . . . . . . 15 ⊢ ((𝑍‘𝑞) = (𝑈‘𝑞) ↔ ¬ (𝑈‘𝑞) ≠ (𝑍‘𝑞))
152	149, 151	sylibr 233	. . . . . . . . . . . . . 14 ⊢ ((((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) ∧ 𝑞 ∈ dom 𝑈) → (𝑍‘𝑞) = (𝑈‘𝑞))
153	143, 152	eqtrd 2764	. . . . . . . . . . . . 13 ⊢ ((((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) ∧ 𝑞 ∈ dom 𝑈) → ((𝑍 ↾ dom 𝑈)‘𝑞) = (𝑈‘𝑞))
154	153	ex 412	. . . . . . . . . . . 12 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → (𝑞 ∈ dom 𝑈 → ((𝑍 ↾ dom 𝑈)‘𝑞) = (𝑈‘𝑞)))
155	141, 154	sylbid 239	. . . . . . . . . . 11 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → (𝑞 ∈ dom (𝑍 ↾ dom 𝑈) → ((𝑍 ↾ dom 𝑈)‘𝑞) = (𝑈‘𝑞)))
156	155	ralrimiv 3137	. . . . . . . . . 10 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → ∀𝑞 ∈ dom (𝑍 ↾ dom 𝑈)((𝑍 ↾ dom 𝑈)‘𝑞) = (𝑈‘𝑞))
157		nofun 27498	. . . . . . . . . . . 12 ⊢ (𝑍 ∈ No → Fun 𝑍)
158		funres 6580	. . . . . . . . . . . 12 ⊢ (Fun 𝑍 → Fun (𝑍 ↾ dom 𝑈))
159	130, 157, 158	3syl 18	. . . . . . . . . . 11 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → Fun (𝑍 ↾ dom 𝑈))
160	131, 94	syl 17	. . . . . . . . . . 11 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → Fun 𝑈)
161		eqfunfv 7027	. . . . . . . . . . 11 ⊢ ((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))))
162	159, 160, 161	syl2anc 583	. . . . . . . . . 10 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {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))))
163	140, 156, 162	mpbir2and 710	. . . . . . . . 9 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → (𝑍 ↾ dom 𝑈) = 𝑈)
164	130, 157	syl 17	. . . . . . . . . . . 12 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → Fun 𝑍)
165		funfn 6568	. . . . . . . . . . . 12 ⊢ (Fun 𝑍 ↔ 𝑍 Fn dom 𝑍)
166	164, 165	sylib 217	. . . . . . . . . . 11 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → 𝑍 Fn dom 𝑍)
167		1oex 8471	. . . . . . . . . . . . . . . . . . 19 ⊢ 1_o ∈ V
168	167	prid1 4758	. . . . . . . . . . . . . . . . . 18 ⊢ 1_o ∈ {1_o, 2_o}
169	168	nosgnn0i 27508	. . . . . . . . . . . . . . . . 17 ⊢ ∅ ≠ 1_o
170	131, 79	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → Ord dom 𝑈)
171		ndmfv 6916	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ dom 𝑈 ∈ dom 𝑈 → (𝑈‘dom 𝑈) = ∅)
172	170, 80, 171	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → (𝑈‘dom 𝑈) = ∅)
173	172	neeq1d 2992	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → ((𝑈‘dom 𝑈) ≠ 1_o ↔ ∅ ≠ 1_o))
174	169, 173	mpbiri 258	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → (𝑈‘dom 𝑈) ≠ 1_o)
175	174	neneqd 2937	. . . . . . . . . . . . . . 15 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → ¬ (𝑈‘dom 𝑈) = 1_o)
176	175	intnanrd 489	. . . . . . . . . . . . . 14 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → ¬ ((𝑈‘dom 𝑈) = 1_o ∧ (𝑍‘dom 𝑈) = ∅))
177	175	intnanrd 489	. . . . . . . . . . . . . 14 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → ¬ ((𝑈‘dom 𝑈) = 1_o ∧ (𝑍‘dom 𝑈) = 2_o))
178		simplr 766	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → 𝑈 <s 𝑍)
179	131, 130, 55	syl2anc 583	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → (𝑈 <s 𝑍 ↔ (𝑈‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}){⟨1_o, ∅⟩, ⟨1_o, 2_o⟩, ⟨∅, 2_o⟩} (𝑍‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)})))
180	178, 179	mpbid 231	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → (𝑈‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}){⟨1_o, ∅⟩, ⟨1_o, 2_o⟩, ⟨∅, 2_o⟩} (𝑍‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}))
181		fveq2 6881	. . . . . . . . . . . . . . . . 17 ⊢ (∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈 → (𝑈‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}) = (𝑈‘dom 𝑈))
182	181	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → (𝑈‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}) = (𝑈‘dom 𝑈))
183		fveq2 6881	. . . . . . . . . . . . . . . . 17 ⊢ (∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈 → (𝑍‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}) = (𝑍‘dom 𝑈))
184	183	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → (𝑍‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}) = (𝑍‘dom 𝑈))
185	180, 182, 184	3brtr3d 5169	. . . . . . . . . . . . . . 15 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → (𝑈‘dom 𝑈){⟨1_o, ∅⟩, ⟨1_o, 2_o⟩, ⟨∅, 2_o⟩} (𝑍‘dom 𝑈))
186		fvex 6894	. . . . . . . . . . . . . . . . 17 ⊢ (𝑈‘dom 𝑈) ∈ V
187		fvex 6894	. . . . . . . . . . . . . . . . 17 ⊢ (𝑍‘dom 𝑈) ∈ V
188	186, 187	brtp 5513	. . . . . . . . . . . . . . . 16 ⊢ ((𝑈‘dom 𝑈){⟨1_o, ∅⟩, ⟨1_o, 2_o⟩, ⟨∅, 2_o⟩} (𝑍‘dom 𝑈) ↔ (((𝑈‘dom 𝑈) = 1_o ∧ (𝑍‘dom 𝑈) = ∅) ∨ ((𝑈‘dom 𝑈) = 1_o ∧ (𝑍‘dom 𝑈) = 2_o) ∨ ((𝑈‘dom 𝑈) = ∅ ∧ (𝑍‘dom 𝑈) = 2_o)))
189		3orrot 1089	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑈‘dom 𝑈) = 1_o ∧ (𝑍‘dom 𝑈) = ∅) ∨ ((𝑈‘dom 𝑈) = 1_o ∧ (𝑍‘dom 𝑈) = 2_o) ∨ ((𝑈‘dom 𝑈) = ∅ ∧ (𝑍‘dom 𝑈) = 2_o)) ↔ (((𝑈‘dom 𝑈) = 1_o ∧ (𝑍‘dom 𝑈) = 2_o) ∨ ((𝑈‘dom 𝑈) = ∅ ∧ (𝑍‘dom 𝑈) = 2_o) ∨ ((𝑈‘dom 𝑈) = 1_o ∧ (𝑍‘dom 𝑈) = ∅)))
190		3orrot 1089	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑈‘dom 𝑈) = 1_o ∧ (𝑍‘dom 𝑈) = 2_o) ∨ ((𝑈‘dom 𝑈) = ∅ ∧ (𝑍‘dom 𝑈) = 2_o) ∨ ((𝑈‘dom 𝑈) = 1_o ∧ (𝑍‘dom 𝑈) = ∅)) ↔ (((𝑈‘dom 𝑈) = ∅ ∧ (𝑍‘dom 𝑈) = 2_o) ∨ ((𝑈‘dom 𝑈) = 1_o ∧ (𝑍‘dom 𝑈) = ∅) ∨ ((𝑈‘dom 𝑈) = 1_o ∧ (𝑍‘dom 𝑈) = 2_o)))
191	188, 189, 190	3bitri 297	. . . . . . . . . . . . . . 15 ⊢ ((𝑈‘dom 𝑈){⟨1_o, ∅⟩, ⟨1_o, 2_o⟩, ⟨∅, 2_o⟩} (𝑍‘dom 𝑈) ↔ (((𝑈‘dom 𝑈) = ∅ ∧ (𝑍‘dom 𝑈) = 2_o) ∨ ((𝑈‘dom 𝑈) = 1_o ∧ (𝑍‘dom 𝑈) = ∅) ∨ ((𝑈‘dom 𝑈) = 1_o ∧ (𝑍‘dom 𝑈) = 2_o)))
192	185, 191	sylib 217	. . . . . . . . . . . . . 14 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → (((𝑈‘dom 𝑈) = ∅ ∧ (𝑍‘dom 𝑈) = 2_o) ∨ ((𝑈‘dom 𝑈) = 1_o ∧ (𝑍‘dom 𝑈) = ∅) ∨ ((𝑈‘dom 𝑈) = 1_o ∧ (𝑍‘dom 𝑈) = 2_o)))
193	176, 177, 192	ecase23d 1469	. . . . . . . . . . . . 13 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → ((𝑈‘dom 𝑈) = ∅ ∧ (𝑍‘dom 𝑈) = 2_o))
194	193	simprd 495	. . . . . . . . . . . 12 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → (𝑍‘dom 𝑈) = 2_o)
195		ndmfv 6916	. . . . . . . . . . . . . 14 ⊢ (¬ dom 𝑈 ∈ dom 𝑍 → (𝑍‘dom 𝑈) = ∅)
196	105	nosgnn0i 27508	. . . . . . . . . . . . . . . 16 ⊢ ∅ ≠ 2_o
197		neeq1 2995	. . . . . . . . . . . . . . . 16 ⊢ ((𝑍‘dom 𝑈) = ∅ → ((𝑍‘dom 𝑈) ≠ 2_o ↔ ∅ ≠ 2_o))
198	196, 197	mpbiri 258	. . . . . . . . . . . . . . 15 ⊢ ((𝑍‘dom 𝑈) = ∅ → (𝑍‘dom 𝑈) ≠ 2_o)
199	198	neneqd 2937	. . . . . . . . . . . . . 14 ⊢ ((𝑍‘dom 𝑈) = ∅ → ¬ (𝑍‘dom 𝑈) = 2_o)
200	195, 199	syl 17	. . . . . . . . . . . . 13 ⊢ (¬ dom 𝑈 ∈ dom 𝑍 → ¬ (𝑍‘dom 𝑈) = 2_o)
201	200	con4i 114	. . . . . . . . . . . 12 ⊢ ((𝑍‘dom 𝑈) = 2_o → dom 𝑈 ∈ dom 𝑍)
202	194, 201	syl 17	. . . . . . . . . . 11 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → dom 𝑈 ∈ dom 𝑍)
203		fnressn 7148	. . . . . . . . . . 11 ⊢ ((𝑍 Fn dom 𝑍 ∧ dom 𝑈 ∈ dom 𝑍) → (𝑍 ↾ {dom 𝑈}) = {⟨dom 𝑈, (𝑍‘dom 𝑈)⟩})
204	166, 202, 203	syl2anc 583	. . . . . . . . . 10 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → (𝑍 ↾ {dom 𝑈}) = {⟨dom 𝑈, (𝑍‘dom 𝑈)⟩})
205	194	opeq2d 4872	. . . . . . . . . . 11 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → ⟨dom 𝑈, (𝑍‘dom 𝑈)⟩ = ⟨dom 𝑈, 2_o⟩)
206	205	sneqd 4632	. . . . . . . . . 10 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → {⟨dom 𝑈, (𝑍‘dom 𝑈)⟩} = {⟨dom 𝑈, 2_o⟩})
207	204, 206	eqtrd 2764	. . . . . . . . 9 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → (𝑍 ↾ {dom 𝑈}) = {⟨dom 𝑈, 2_o⟩})
208	163, 207	uneq12d 4156	. . . . . . . 8 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → ((𝑍 ↾ dom 𝑈) ∪ (𝑍 ↾ {dom 𝑈})) = (𝑈 ∪ {⟨dom 𝑈, 2_o⟩}))
209	124, 208	eqtrid 2776	. . . . . . 7 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → (𝑍 ↾ suc dom 𝑈) = (𝑈 ∪ {⟨dom 𝑈, 2_o⟩}))
210		sonr 5601	. . . . . . . . 9 ⊢ (( <s Or No ∧ (𝑈 ∪ {⟨dom 𝑈, 2_o⟩}) ∈ No ) → ¬ (𝑈 ∪ {⟨dom 𝑈, 2_o⟩}) <s (𝑈 ∪ {⟨dom 𝑈, 2_o⟩}))
211	12, 210	mpan 687	. . . . . . . 8 ⊢ ((𝑈 ∪ {⟨dom 𝑈, 2_o⟩}) ∈ No → ¬ (𝑈 ∪ {⟨dom 𝑈, 2_o⟩}) <s (𝑈 ∪ {⟨dom 𝑈, 2_o⟩}))
212	131, 106, 211	3syl 18	. . . . . . 7 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → ¬ (𝑈 ∪ {⟨dom 𝑈, 2_o⟩}) <s (𝑈 ∪ {⟨dom 𝑈, 2_o⟩}))
213	209, 212	eqnbrtrd 5156	. . . . . 6 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → ¬ (𝑍 ↾ suc dom 𝑈) <s (𝑈 ∪ {⟨dom 𝑈, 2_o⟩}))
214	120, 213	jaodan 954	. . . . 5 ⊢ (((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) ∧ (∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈 ∨ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈)) → ¬ (𝑍 ↾ suc dom 𝑈) <s (𝑈 ∪ {⟨dom 𝑈, 2_o⟩}))
215	214	ex 412	. . . 4 ⊢ ((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) → ((∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ dom 𝑈 ∨ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈) → ¬ (𝑍 ↾ suc dom 𝑈) <s (𝑈 ∪ {⟨dom 𝑈, 2_o⟩})))
216	29, 215	sylbid 239	. . 3 ⊢ ((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) → (∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ⊆ dom 𝑈 → ¬ (𝑍 ↾ suc dom 𝑈) <s (𝑈 ∪ {⟨dom 𝑈, 2_o⟩})))
217	23, 216	mpd 15	. 2 ⊢ ((((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) ∧ 𝑈 <s 𝑍) → ¬ (𝑍 ↾ suc dom 𝑈) <s (𝑈 ∪ {⟨dom 𝑈, 2_o⟩}))
218	5, 217	mpdan 684	1 ⊢ (((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) → ¬ (𝑍 ↾ suc dom 𝑈) <s (𝑈 ∪ {⟨dom 𝑈, 2_o⟩}))

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 2932 ∀wral 3053 ∃wrex 3062 {crab 3424 Vcvv 3466 ∪ cun 3938 ∩ cin 3939 ⊆ wss 3940 ∅c0 4314 {csn 4620 {ctp 4624 ⟨cop 4626 ∩ cint 4940 class class class wbr 5138 Or wor 5577 × cxp 5664 dom cdm 5666 ↾ cres 5668 Rel wrel 5671 Ord word 6353 Oncon0 6354 suc csuc 6356 Fun wfun 6527 Fn wfn 6528 ‘cfv 6533 1_oc1o 8454 2_oc2o 8455 No csur 27489 <s cslt 27490

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 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pr 5417 ax-un 7718

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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-ord 6357 df-on 6358 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-1o 8461 df-2o 8462 df-no 27492 df-slt 27493

This theorem is referenced by: nosupbnd2 27565

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