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Theorem nosupbnd2lem1 28072

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 1216	. . 3 ⊢ (((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) → 𝑈 ∈ 𝐴)
2		simp3 1156	. . 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 5106	. . . 4 ⊢ (𝑎 = 𝑈 → (𝑎 <s 𝑍 ↔ 𝑈 <s 𝑍))
4	3	rspcv 3573	. . 3 ⊢ (𝑈 ∈ 𝐴 → (∀𝑎 ∈ 𝐴 𝑎 <s 𝑍 → 𝑈 <s 𝑍))
5	1, 2, 4	sylc 66	. 2 ⊢ (((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) → 𝑈 <s 𝑍)
6		simpl21 1270	. . . . 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 1243	. . . . 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 3932	. . . 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 1272	. . . 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 1225	. . . . . . . . . 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 3932	. . . . . . . . 9 ⊢ (((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) → 𝑈 ∈ No)
12		ltsso 28033	. . . . . . . . . 10 ⊢ <s Or No
13		sonr 5583	. . . . . . . . . 10 ⊢ (( <s Or No ∧ 𝑈 ∈ No) → ¬ 𝑈 <s 𝑈)
14	12, 13	mpan 703	. . . . . . . . 9 ⊢ (𝑈 ∈ No → ¬ 𝑈 <s 𝑈)
15	11, 14	syl 18	. . . . . . . 8 ⊢ (((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) → ¬ 𝑈 <s 𝑈)
16		breq2 5107	. . . . . . . . 9 ⊢ (𝑈 = 𝑍 → (𝑈 <s 𝑈 ↔ 𝑈 <s 𝑍))
17	16	notbid 321	. . . . . . . 8 ⊢ (𝑈 = 𝑍 → (¬ 𝑈 <s 𝑈 ↔ ¬ 𝑈 <s 𝑍))
18	15, 17	syl5ibcom 248	. . . . . . 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 135	. . . . . 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 412	. . . . 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 2963	. . . 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 28043	. . . 4 ⊢ ((𝑈 ∈ No ∧ 𝑍 ∈ No ∧ 𝑈 ≠ 𝑍) → ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ⊆ dom 𝑈)
23	8, 9, 21, 22	syl3anc 1398	. . 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 28022	. . . . . 6 ⊢ ((𝑈 ∈ No ∧ 𝑍 ∈ No ∧ 𝑈 ≠ 𝑍) → ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ On)
25	8, 9, 21, 24	syl3anc 1398	. . . . 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 28007	. . . . . 6 ⊢ (𝑈 ∈ No → dom 𝑈 ∈ On)
27	8, 26	syl 18	. . . . 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 6404	. . . . 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 596	. . . 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 486	. . . . . . . . . . . 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 486	. . . . . . . . . . . 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 486	. . . . . . . . . . . 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 1398	. . . . . . . . . . 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 6387	. . . . . . . . . . . . . . . 16 ⊢ ((∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} ∈ On ∧ 𝑞 ∈ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}) → 𝑞 ∈ On)
35	33, 34	sylan 592	. . . . . . . . . . . . . . 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 490	. . . . . . . . . . . . . . 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 6885	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑞 → (𝑈‘𝑥) = (𝑈‘𝑞))
38		fveq2 6885	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑞 → (𝑍‘𝑥) = (𝑍‘𝑞))
39	37, 38	neeq12d 3017	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑞 → ((𝑈‘𝑥) ≠ (𝑍‘𝑥) ↔ (𝑈‘𝑞) ≠ (𝑍‘𝑞)))
40	39	onnminsb 7813	. . . . . . . . . . . . . . 15 ⊢ (𝑞 ∈ On → (𝑞 ∈ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} → ¬ (𝑈‘𝑞) ≠ (𝑍‘𝑞)))
41	35, 36, 40	sylc 66	. . . . . . . . . . . . . 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 2957	. . . . . . . . . . . . . . 15 ⊢ ((𝑈‘𝑞) ≠ (𝑍‘𝑞) ↔ ¬ (𝑈‘𝑞) = (𝑍‘𝑞))
43	42	con2bii 360	. . . . . . . . . . . . . 14 ⊢ ((𝑈‘𝑞) = (𝑍‘𝑞) ↔ ¬ (𝑈‘𝑞) ≠ (𝑍‘𝑞))
44	41, 43	sylibr 237	. . . . . . . . . . . . 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 781	. . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . 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 6410	. . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . 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 712	. . . . . . . . . . . . . 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 6905	. . . . . . . . . . . . 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 2799	. . . . . . . . . . . 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 3155	. . . . . . . . . . 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 781	. . . . . . . . . . . . 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		ltsval2 28013	. . . . . . . . . . . . . 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 596	. . . . . . . . . . . . 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 235	. . . . . . . . . . . 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 490	. . . . . . . . . . . . 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 6905	. . . . . . . . . . . 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 5133	. . . . . . . . . . 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 3317	. . . . . . . . . . . . 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 6885	. . . . . . . . . . . . . 14 ⊢ (𝑝 = ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} → (𝑈‘𝑝) = (𝑈‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}))
63		fveq2 6885	. . . . . . . . . . . . . 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 5116	. . . . . . . . . . . . 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 644	. . . . . . . . . . . 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 3577	. . . . . . . . . . 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 850	. . . . . . . . . 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 28017	. . . . . . . . . . . 12 ⊢ ((𝑍 ∈ No ∧ dom 𝑈 ∈ On) → (𝑍 ↾ dom 𝑈) ∈ No)
69	31, 46, 68	syl2anc 596	. . . . . . . . . . 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		ltsval 28004	. . . . . . . . . . 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 596	. . . . . . . . . 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 260	. . . . . . . . 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 5663	. . . . . . . . . . . . 13 ⊢ ({⟨dom 𝑈, 2_o⟩} ↾ dom 𝑈) = ({⟨dom 𝑈, 2_o⟩} ∩ (dom 𝑈 × V))
74		2on 8490	. . . . . . . . . . . . . . . 16 ⊢ 2_o ∈ On
75		xpsng 7140	. . . . . . . . . . . . . . . 16 ⊢ ((dom 𝑈 ∈ On ∧ 2_o ∈ On) → ({dom 𝑈} × {2_o}) = {⟨dom 𝑈, 2_o⟩})
76	46, 74, 75	sylancl 598	. . . . . . . . . . . . . . 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 4165	. . . . . . . . . . . . . 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 4155	. . . . . . . . . . . . . . . 16 ⊢ ({dom 𝑈} ∩ dom 𝑈) = (dom 𝑈 ∩ {dom 𝑈})
79		nodmord 28010	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑈 ∈ No → Ord dom 𝑈)
80		ordirr 6380	. . . . . . . . . . . . . . . . . 18 ⊢ (Ord dom 𝑈 → ¬ dom 𝑈 ∈ dom 𝑈)
81	30, 79, 80	3syl 19	. . . . . . . . . . . . . . . . 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 4672	. . . . . . . . . . . . . . . . 17 ⊢ ((dom 𝑈 ∩ {dom 𝑈}) = ∅ ↔ ¬ dom 𝑈 ∈ dom 𝑈)
83	81, 82	sylibr 237	. . . . . . . . . . . . . . . 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 2808	. . . . . . . . . . . . . . 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 6152	. . . . . . . . . . . . . . 15 ⊢ (({dom 𝑈} ∩ dom 𝑈) = ∅ → (({dom 𝑈} × {2_o}) ∩ (dom 𝑈 × V)) = ∅)
86	84, 85	syl 18	. . . . . . . . . . . . . 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 2798	. . . . . . . . . . . . 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 2808	. . . . . . . . . . . 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 4115	. . . . . . . . . . 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 5985	. . . . . . . . . . 11 ⊢ ((𝑈 ∪ {⟨dom 𝑈, 2_o⟩}) ↾ dom 𝑈) = ((𝑈 ↾ dom 𝑈) ∪ ({⟨dom 𝑈, 2_o⟩} ↾ dom 𝑈))
91		un0 4344	. . . . . . . . . . . 12 ⊢ ((𝑈 ↾ dom 𝑈) ∪ ∅) = (𝑈 ↾ dom 𝑈)
92	91	eqcomi 2770	. . . . . . . . . . 11 ⊢ (𝑈 ↾ dom 𝑈) = ((𝑈 ↾ dom 𝑈) ∪ ∅)
93	89, 90, 92	3eqtr4g 2821	. . . . . . . . . 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 28006	. . . . . . . . . . 11 ⊢ (𝑈 ∈ No → Fun 𝑈)
95		funrel 6556	. . . . . . . . . . 11 ⊢ (Fun 𝑈 → Rel 𝑈)
96		resdm 6015	. . . . . . . . . . 11 ⊢ (Rel 𝑈 → (𝑈 ↾ dom 𝑈) = 𝑈)
97	30, 94, 95, 96	4syl 20	. . . . . . . . . 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)
98	93, 97	eqtrd 2796	. . . . . . . . 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)
99		sssucid 6445	. . . . . . . . . 10 ⊢ dom 𝑈 ⊆ suc dom 𝑈
100		resabs1 5997	. . . . . . . . . 10 ⊢ (dom 𝑈 ⊆ suc dom 𝑈 → ((𝑍 ↾ suc dom 𝑈) ↾ dom 𝑈) = (𝑍 ↾ dom 𝑈))
101	99, 100	mp1i 14	. . . . . . . . 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))
102	72, 98, 101	3brtr4d 5137	. . . . . . . 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))
103	74	elexi 3473	. . . . . . . . . . . . 13 ⊢ 2_o ∈ V
104	103	prid2 4724	. . . . . . . . . . . 12 ⊢ 2_o ∈ {1_o, 2_o}
105	104	noextend 28023	. . . . . . . . . . 11 ⊢ (𝑈 ∈ No → (𝑈 ∪ {⟨dom 𝑈, 2_o⟩}) ∈ No)
106	8, 105	syl 18	. . . . . . . . . 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)
107	106	adantr 486	. . . . . . . . 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)
108		onsucb 7828	. . . . . . . . . . . 12 ⊢ (dom 𝑈 ∈ On ↔ suc dom 𝑈 ∈ On)
109	27, 108	sylib 221	. . . . . . . . . . 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)
110		noreson 28017	. . . . . . . . . . 11 ⊢ ((𝑍 ∈ No ∧ suc dom 𝑈 ∈ On) → (𝑍 ↾ suc dom 𝑈) ∈ No)
111	9, 109, 110	syl2anc 596	. . . . . . . . . 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)
112	111	adantr 486	. . . . . . . . 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)
113		ltsres 28019	. . . . . . . . 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)))
114	107, 112, 46, 113	syl3anc 1398	. . . . . . . 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)))
115	102, 114	mpd 16	. . . . . . 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))
116		soasym 5592	. . . . . . . . 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⟩})))
117	12, 116	mpan 703	. . . . . . . 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⟩})))
118	107, 112, 117	syl2anc 596	. . . . . . 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⟩})))
119	115, 118	mpd 16	. . . . . 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⟩}))
120		df-suc 6368	. . . . . . . . . 10 ⊢ suc dom 𝑈 = (dom 𝑈 ∪ {dom 𝑈})
121	120	reseq2i 5967	. . . . . . . . 9 ⊢ (𝑍 ↾ suc dom 𝑈) = (𝑍 ↾ (dom 𝑈 ∪ {dom 𝑈}))
122		resundi 5984	. . . . . . . . 9 ⊢ (𝑍 ↾ (dom 𝑈 ∪ {dom 𝑈})) = ((𝑍 ↾ dom 𝑈) ∪ (𝑍 ↾ {dom 𝑈}))
123	121, 122	eqtri 2784	. . . . . . . 8 ⊢ (𝑍 ↾ suc dom 𝑈) = ((𝑍 ↾ dom 𝑈) ∪ (𝑍 ↾ {dom 𝑈}))
124		dmres 6003	. . . . . . . . . . 11 ⊢ dom (𝑍 ↾ dom 𝑈) = (dom 𝑈 ∩ dom 𝑍)
125		simpr 490	. . . . . . . . . . . . 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)
126		necom 3009	. . . . . . . . . . . . . . . 16 ⊢ ((𝑈‘𝑥) ≠ (𝑍‘𝑥) ↔ (𝑍‘𝑥) ≠ (𝑈‘𝑥))
127	126	rabbii 3418	. . . . . . . . . . . . . . 15 ⊢ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = {𝑥 ∈ On ∣ (𝑍‘𝑥) ≠ (𝑈‘𝑥)}
128	127	inteqi 4911	. . . . . . . . . . . . . 14 ⊢ ∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = ∩ {𝑥 ∈ On ∣ (𝑍‘𝑥) ≠ (𝑈‘𝑥)}
129	9	adantr 486	. . . . . . . . . . . . . . 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)
130	8	adantr 486	. . . . . . . . . . . . . . 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)
131	21	adantr 486	. . . . . . . . . . . . . . . 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)
132	131	necomd 3011	. . . . . . . . . . . . . . 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)
133		nosepssdm 28043	. . . . . . . . . . . . . . 15 ⊢ ((𝑍 ∈ No ∧ 𝑈 ∈ No ∧ 𝑍 ≠ 𝑈) → ∩ {𝑥 ∈ On ∣ (𝑍‘𝑥) ≠ (𝑈‘𝑥)} ⊆ dom 𝑍)
134	129, 130, 132, 133	syl3anc 1398	. . . . . . . . . . . . . 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)
135	128, 134	eqsstrid 3969	. . . . . . . . . . . . 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)
136	125, 135	eqsstrrd 3966	. . . . . . . . . . . 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)
137		dfss2 3917	. . . . . . . . . . . 12 ⊢ (dom 𝑈 ⊆ dom 𝑍 ↔ (dom 𝑈 ∩ dom 𝑍) = dom 𝑈)
138	136, 137	sylib 221	. . . . . . . . . . 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)
139	124, 138	eqtrid 2808	. . . . . . . . . 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)
140	139	eleq2d 2847	. . . . . . . . . . . 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))
141		simpr 490	. . . . . . . . . . . . . . 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)
142	141	fvresd 6905	. . . . . . . . . . . . . 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))
143	130, 26	syl 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 ∈ On)
144		onelon 6387	. . . . . . . . . . . . . . . . 17 ⊢ ((dom 𝑈 ∈ On ∧ 𝑞 ∈ dom 𝑈) → 𝑞 ∈ On)
145	143, 144	sylan 592	. . . . . . . . . . . . . . . 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)
146	125	eleq2d 2847	. . . . . . . . . . . . . . . . 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))
147	146	biimpar 483	. . . . . . . . . . . . . . . 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)})
148	145, 147, 40	sylc 66	. . . . . . . . . . . . . . 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))
149		nesym 3012	. . . . . . . . . . . . . . . 16 ⊢ ((𝑈‘𝑞) ≠ (𝑍‘𝑞) ↔ ¬ (𝑍‘𝑞) = (𝑈‘𝑞))
150	149	con2bii 360	. . . . . . . . . . . . . . 15 ⊢ ((𝑍‘𝑞) = (𝑈‘𝑞) ↔ ¬ (𝑈‘𝑞) ≠ (𝑍‘𝑞))
151	148, 150	sylibr 237	. . . . . . . . . . . . . 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))
152	142, 151	eqtrd 2796	. . . . . . . . . . . . 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))
153	152	ex 418	. . . . . . . . . . . 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)))
154	140, 153	sylbid 243	. . . . . . . . . . 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)))
155	154	ralrimiv 3154	. . . . . . . . . 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))
156		nofun 28006	. . . . . . . . . . . 12 ⊢ (𝑍 ∈ No → Fun 𝑍)
157		funres 6582	. . . . . . . . . . . 12 ⊢ (Fun 𝑍 → Fun (𝑍 ↾ dom 𝑈))
158	129, 156, 157	3syl 19	. . . . . . . . . . 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))
159	130, 94	syl 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 𝑈)
160		eqfunfv 7035	. . . . . . . . . . 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))))
161	158, 159, 160	syl2anc 596	. . . . . . . . . 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))))
162	139, 155, 161	mpbir2and 726	. . . . . . . . 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)
163	129, 156	syl 18	. . . . . . . . . . . 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)
164		funfn 6570	. . . . . . . . . . . 12 ⊢ (Fun 𝑍 ↔ 𝑍 Fn dom 𝑍)
165	163, 164	sylib 221	. . . . . . . . . . 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)
166		1oex 8486	. . . . . . . . . . . . . . . . . . 19 ⊢ 1_o ∈ V
167	166	prid1 4723	. . . . . . . . . . . . . . . . . 18 ⊢ 1_o ∈ {1_o, 2_o}
168	167	nosgnn0i 28016	. . . . . . . . . . . . . . . . 17 ⊢ ∅ ≠ 1_o
169		ndmfv 6917	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ dom 𝑈 ∈ dom 𝑈 → (𝑈‘dom 𝑈) = ∅)
170	130, 79, 80, 169	4syl 20	. . . . . . . . . . . . . . . . . 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) = ∅)
171	170	neeq1d 3015	. . . . . . . . . . . . . . . . 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))
172	168, 171	mpbiri 261	. . . . . . . . . . . . . . . 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)
173	172	neneqd 2961	. . . . . . . . . . . . . . 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)
174	173	intnanrd 495	. . . . . . . . . . . . . 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) = ∅))
175	173	intnanrd 495	. . . . . . . . . . . . . 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))
176		simplr 781	. . . . . . . . . . . . . . . . 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)
177	130, 129, 55	syl2anc 596	. . . . . . . . . . . . . . . . 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)})))
178	176, 177	mpbid 235	. . . . . . . . . . . . . . . 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)}))
179		fveq2 6885	. . . . . . . . . . . . . . . . 17 ⊢ (∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈 → (𝑈‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}) = (𝑈‘dom 𝑈))
180	179	adantl 487	. . . . . . . . . . . . . . . 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))
181		fveq2 6885	. . . . . . . . . . . . . . . . 17 ⊢ (∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)} = dom 𝑈 → (𝑍‘∩ {𝑥 ∈ On ∣ (𝑈‘𝑥) ≠ (𝑍‘𝑥)}) = (𝑍‘dom 𝑈))
182	181	adantl 487	. . . . . . . . . . . . . . . 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))
183	178, 180, 182	3brtr3d 5136	. . . . . . . . . . . . . . 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))
184		fvex 6898	. . . . . . . . . . . . . . . . 17 ⊢ (𝑈‘dom 𝑈) ∈ V
185		fvex 6898	. . . . . . . . . . . . . . . . 17 ⊢ (𝑍‘dom 𝑈) ∈ V
186	184, 185	brtp 5497	. . . . . . . . . . . . . . . 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)))
187		3orrot 1108	. . . . . . . . . . . . . . . 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) = ∅)))
188		3orrot 1108	. . . . . . . . . . . . . . . 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)))
189	186, 187, 188	3bitri 300	. . . . . . . . . . . . . . 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)))
190	183, 189	sylib 221	. . . . . . . . . . . . . 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)))
191	174, 175, 190	ecase23d 1503	. . . . . . . . . . . . 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))
192	191	simprd 501	. . . . . . . . . . . 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)
193		ndmfv 6917	. . . . . . . . . . . . . 14 ⊢ (¬ dom 𝑈 ∈ dom 𝑍 → (𝑍‘dom 𝑈) = ∅)
194	104	nosgnn0i 28016	. . . . . . . . . . . . . . . 16 ⊢ ∅ ≠ 2_o
195		neeq1 3018	. . . . . . . . . . . . . . . 16 ⊢ ((𝑍‘dom 𝑈) = ∅ → ((𝑍‘dom 𝑈) ≠ 2_o ↔ ∅ ≠ 2_o))
196	194, 195	mpbiri 261	. . . . . . . . . . . . . . 15 ⊢ ((𝑍‘dom 𝑈) = ∅ → (𝑍‘dom 𝑈) ≠ 2_o)
197	196	neneqd 2961	. . . . . . . . . . . . . 14 ⊢ ((𝑍‘dom 𝑈) = ∅ → ¬ (𝑍‘dom 𝑈) = 2_o)
198	193, 197	syl 18	. . . . . . . . . . . . 13 ⊢ (¬ dom 𝑈 ∈ dom 𝑍 → ¬ (𝑍‘dom 𝑈) = 2_o)
199	198	con4i 115	. . . . . . . . . . . 12 ⊢ ((𝑍‘dom 𝑈) = 2_o → dom 𝑈 ∈ dom 𝑍)
200	192, 199	syl 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) → dom 𝑈 ∈ dom 𝑍)
201		fnressn 7162	. . . . . . . . . . 11 ⊢ ((𝑍 Fn dom 𝑍 ∧ dom 𝑈 ∈ dom 𝑍) → (𝑍 ↾ {dom 𝑈}) = {⟨dom 𝑈, (𝑍‘dom 𝑈)⟩})
202	165, 200, 201	syl2anc 596	. . . . . . . . . 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)⟩})
203	192	opeq2d 4840	. . . . . . . . . . 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⟩)
204	203	sneqd 4596	. . . . . . . . . 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⟩})
205	202, 204	eqtrd 2796	. . . . . . . . 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⟩})
206	162, 205	uneq12d 4116	. . . . . . . 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⟩}))
207	123, 206	eqtrid 2808	. . . . . . 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⟩}))
208		sonr 5583	. . . . . . . . 9 ⊢ (( <s Or No ∧ (𝑈 ∪ {⟨dom 𝑈, 2_o⟩}) ∈ No) → ¬ (𝑈 ∪ {⟨dom 𝑈, 2_o⟩}) <s (𝑈 ∪ {⟨dom 𝑈, 2_o⟩}))
209	12, 208	mpan 703	. . . . . . . 8 ⊢ ((𝑈 ∪ {⟨dom 𝑈, 2_o⟩}) ∈ No → ¬ (𝑈 ∪ {⟨dom 𝑈, 2_o⟩}) <s (𝑈 ∪ {⟨dom 𝑈, 2_o⟩}))
210	130, 105, 209	3syl 19	. . . . . . 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⟩}))
211	207, 210	eqnbrtrd 5123	. . . . . 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⟩}))
212	119, 211	jaodan 972	. . . . 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⟩}))
213	212	ex 418	. . . 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⟩})))
214	29, 213	sylbid 243	. . 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⟩})))
215	23, 214	mpd 16	. 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⟩}))
216	5, 215	mpdan 700	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

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 ∨ w3o 1102 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2956 ∀wral 3077 ∃wrex 3087 {crab 3413 Vcvv 3451 ∪ cun 3897 ∩ cin 3898 ⊆ wss 3899 ∅c0 4279 {csn 4584 {ctp 4588 ⟨cop 4590 ∩ cint 4907 class class class wbr 5103 Or wor 5558 × cxp 5649 dom cdm 5651 ↾ cres 5653 Rel wrel 5656 Ord word 6361 Oncon0 6362 suc csuc 6364 Fun wfun 6532 Fn wfn 6533 ‘cfv 6538 1_oc1o 8469 2_oc2o 8470 Nocsur 27997 <s clts 27998

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7751

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3078 df-rex 3088 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-ord 6365 df-on 6366 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-1o 8476 df-2o 8477 df-no 28000 df-lts 28001

This theorem is used by: nosupbnd2 28073

W3C validator