Theorem sltres 31061

Description: If the restrictions of two surreals to a given ordinal obey surreal less than, then so do the two surreals themselves. (Contributed by Scott Fenton, 4-Sep-2011.)

Assertion

Ref	Expression
sltres	⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) → ((𝐴 ↾ 𝑋) <s (𝐵 ↾ 𝑋) → 𝐴 <s 𝐵))

Proof of Theorem sltres

Dummy variables 𝑎 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		noreson 31057	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑋 ∈ On) → (𝐴 ↾ 𝑋) ∈ No )
2	1	3adant2 1073	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) → (𝐴 ↾ 𝑋) ∈ No )
3		noreson 31057	. . . . . . 7 ⊢ ((𝐵 ∈ No ∧ 𝑋 ∈ On) → (𝐵 ↾ 𝑋) ∈ No )
4	3	3adant1 1072	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) → (𝐵 ↾ 𝑋) ∈ No )
5		sltintdifex 31060	. . . . . . 7 ⊢ (((𝐴 ↾ 𝑋) ∈ No ∧ (𝐵 ↾ 𝑋) ∈ No ) → ((𝐴 ↾ 𝑋) <s (𝐵 ↾ 𝑋) → ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ V))
6		onintrab 6893	. . . . . . 7 ⊢ (∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ V ↔ ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ On)
7	5, 6	syl6ib 240	. . . . . 6 ⊢ (((𝐴 ↾ 𝑋) ∈ No ∧ (𝐵 ↾ 𝑋) ∈ No ) → ((𝐴 ↾ 𝑋) <s (𝐵 ↾ 𝑋) → ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ On))
8	2, 4, 7	syl2anc 691	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) → ((𝐴 ↾ 𝑋) <s (𝐵 ↾ 𝑋) → ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ On))
9	8	imp 444	. . . 4 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ (𝐴 ↾ 𝑋) <s (𝐵 ↾ 𝑋)) → ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ On)
10		simpl3 1059	. . . . . . . 8 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ (𝐴 ↾ 𝑋) <s (𝐵 ↾ 𝑋)) → 𝑋 ∈ On)
11		sltval2 31053	. . . . . . . . . . . 12 ⊢ (((𝐴 ↾ 𝑋) ∈ No ∧ (𝐵 ↾ 𝑋) ∈ No ) → ((𝐴 ↾ 𝑋) <s (𝐵 ↾ 𝑋) ↔ ((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)})))
12	2, 4, 11	syl2anc 691	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) → ((𝐴 ↾ 𝑋) <s (𝐵 ↾ 𝑋) ↔ ((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)})))
13		fvex 6113	. . . . . . . . . . . . 13 ⊢ ((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) ∈ V
14		fvex 6113	. . . . . . . . . . . . 13 ⊢ ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) ∈ V
15	13, 14	brtp 30892	. . . . . . . . . . . 12 ⊢ (((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) ↔ ((((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅) ∨ (((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 2𝑜) ∨ (((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅ ∧ ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 2𝑜)))
16		1n0 7462	. . . . . . . . . . . . . . . . . 18 ⊢ 1𝑜 ≠ ∅
17	16	neii 2784	. . . . . . . . . . . . . . . . 17 ⊢ ¬ 1𝑜 = ∅
18		eqeq1 2614	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 1𝑜 → (((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅ ↔ 1𝑜 = ∅))
19	17, 18	mtbiri 316	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 1𝑜 → ¬ ((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅)
20		ndmfv 6128	. . . . . . . . . . . . . . . 16 ⊢ (¬ ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom (𝐴 ↾ 𝑋) → ((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅)
21	19, 20	nsyl2 141	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 1𝑜 → ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom (𝐴 ↾ 𝑋))
22	21	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅) → ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom (𝐴 ↾ 𝑋))
23	22	orcd 406	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅) → (∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom (𝐴 ↾ 𝑋) ∨ ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom (𝐵 ↾ 𝑋)))
24	21	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 2𝑜) → ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom (𝐴 ↾ 𝑋))
25	24	orcd 406	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 2𝑜) → (∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom (𝐴 ↾ 𝑋) ∨ ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom (𝐵 ↾ 𝑋)))
26		2on 7455	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 2𝑜 ∈ On
27	26	elexi 3186	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2𝑜 ∈ V
28	27	prid2 4242	. . . . . . . . . . . . . . . . . . 19 ⊢ 2𝑜 ∈ {1𝑜, 2𝑜}
29	28	nosgnn0i 31056	. . . . . . . . . . . . . . . . . 18 ⊢ ∅ ≠ 2𝑜
30	29	neii 2784	. . . . . . . . . . . . . . . . 17 ⊢ ¬ ∅ = 2𝑜
31		eqeq1 2614	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 2𝑜 → (((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅ ↔ 2𝑜 = ∅))
32		eqcom 2617	. . . . . . . . . . . . . . . . . 18 ⊢ (2𝑜 = ∅ ↔ ∅ = 2𝑜)
33	31, 32	syl6bb 275	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 2𝑜 → (((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅ ↔ ∅ = 2𝑜))
34	30, 33	mtbiri 316	. . . . . . . . . . . . . . . 16 ⊢ (((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 2𝑜 → ¬ ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅)
35		ndmfv 6128	. . . . . . . . . . . . . . . 16 ⊢ (¬ ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom (𝐵 ↾ 𝑋) → ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅)
36	34, 35	nsyl2 141	. . . . . . . . . . . . . . 15 ⊢ (((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 2𝑜 → ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom (𝐵 ↾ 𝑋))
37	36	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅ ∧ ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 2𝑜) → ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom (𝐵 ↾ 𝑋))
38	37	olcd 407	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅ ∧ ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 2𝑜) → (∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom (𝐴 ↾ 𝑋) ∨ ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom (𝐵 ↾ 𝑋)))
39	23, 25, 38	3jaoi 1383	. . . . . . . . . . . 12 ⊢ (((((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅) ∨ (((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 2𝑜) ∨ (((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅ ∧ ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 2𝑜)) → (∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom (𝐴 ↾ 𝑋) ∨ ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom (𝐵 ↾ 𝑋)))
40	15, 39	sylbi 206	. . . . . . . . . . 11 ⊢ (((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) → (∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom (𝐴 ↾ 𝑋) ∨ ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom (𝐵 ↾ 𝑋)))
41	12, 40	syl6bi 242	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) → ((𝐴 ↾ 𝑋) <s (𝐵 ↾ 𝑋) → (∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom (𝐴 ↾ 𝑋) ∨ ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom (𝐵 ↾ 𝑋))))
42	41	imp 444	. . . . . . . . 9 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ (𝐴 ↾ 𝑋) <s (𝐵 ↾ 𝑋)) → (∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom (𝐴 ↾ 𝑋) ∨ ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom (𝐵 ↾ 𝑋)))
43		dmres 5339	. . . . . . . . . . . 12 ⊢ dom (𝐴 ↾ 𝑋) = (𝑋 ∩ dom 𝐴)
44	43	elin2 3763	. . . . . . . . . . 11 ⊢ (∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom (𝐴 ↾ 𝑋) ↔ (∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ 𝑋 ∧ ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom 𝐴))
45	44	simplbi 475	. . . . . . . . . 10 ⊢ (∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom (𝐴 ↾ 𝑋) → ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ 𝑋)
46		dmres 5339	. . . . . . . . . . . 12 ⊢ dom (𝐵 ↾ 𝑋) = (𝑋 ∩ dom 𝐵)
47	46	elin2 3763	. . . . . . . . . . 11 ⊢ (∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom (𝐵 ↾ 𝑋) ↔ (∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ 𝑋 ∧ ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom 𝐵))
48	47	simplbi 475	. . . . . . . . . 10 ⊢ (∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom (𝐵 ↾ 𝑋) → ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ 𝑋)
49	45, 48	jaoi 393	. . . . . . . . 9 ⊢ ((∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom (𝐴 ↾ 𝑋) ∨ ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom (𝐵 ↾ 𝑋)) → ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ 𝑋)
50	42, 49	syl 17	. . . . . . . 8 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ (𝐴 ↾ 𝑋) <s (𝐵 ↾ 𝑋)) → ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ 𝑋)
51		onelss 5683	. . . . . . . 8 ⊢ (𝑋 ∈ On → (∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ 𝑋 → ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ⊆ 𝑋))
52	10, 50, 51	sylc 63	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ (𝐴 ↾ 𝑋) <s (𝐵 ↾ 𝑋)) → ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ⊆ 𝑋)
53	52	sselda 3568	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ (𝐴 ↾ 𝑋) <s (𝐵 ↾ 𝑋)) ∧ 𝑦 ∈ ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) → 𝑦 ∈ 𝑋)
54		onelon 5665	. . . . . . . . 9 ⊢ ((∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ On ∧ 𝑦 ∈ ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) → 𝑦 ∈ On)
55	9, 54	sylan 487	. . . . . . . 8 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ (𝐴 ↾ 𝑋) <s (𝐵 ↾ 𝑋)) ∧ 𝑦 ∈ ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) → 𝑦 ∈ On)
56		intss1 4427	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} → ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ⊆ 𝑦)
57		ontri1 5674	. . . . . . . . . . . . 13 ⊢ ((∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ On ∧ 𝑦 ∈ On) → (∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ⊆ 𝑦 ↔ ¬ 𝑦 ∈ ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}))
58	56, 57	syl5ib 233	. . . . . . . . . . . 12 ⊢ ((∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ On ∧ 𝑦 ∈ On) → (𝑦 ∈ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} → ¬ 𝑦 ∈ ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}))
59	58	con2d 128	. . . . . . . . . . 11 ⊢ ((∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ On ∧ 𝑦 ∈ On) → (𝑦 ∈ ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} → ¬ 𝑦 ∈ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}))
60	9, 59	sylan 487	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ (𝐴 ↾ 𝑋) <s (𝐵 ↾ 𝑋)) ∧ 𝑦 ∈ On) → (𝑦 ∈ ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} → ¬ 𝑦 ∈ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}))
61	60	impancom 455	. . . . . . . . 9 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ (𝐴 ↾ 𝑋) <s (𝐵 ↾ 𝑋)) ∧ 𝑦 ∈ ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) → (𝑦 ∈ On → ¬ 𝑦 ∈ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}))
62	55, 61	mpd 15	. . . . . . . 8 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ (𝐴 ↾ 𝑋) <s (𝐵 ↾ 𝑋)) ∧ 𝑦 ∈ ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) → ¬ 𝑦 ∈ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)})
63		fveq2 6103	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑦 → ((𝐴 ↾ 𝑋)‘𝑎) = ((𝐴 ↾ 𝑋)‘𝑦))
64		fveq2 6103	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑦 → ((𝐵 ↾ 𝑋)‘𝑎) = ((𝐵 ↾ 𝑋)‘𝑦))
65	63, 64	neeq12d 2843	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑦 → (((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎) ↔ ((𝐴 ↾ 𝑋)‘𝑦) ≠ ((𝐵 ↾ 𝑋)‘𝑦)))
66	65	elrab 3331	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ↔ (𝑦 ∈ On ∧ ((𝐴 ↾ 𝑋)‘𝑦) ≠ ((𝐵 ↾ 𝑋)‘𝑦)))
67	66	simplbi2 653	. . . . . . . . 9 ⊢ (𝑦 ∈ On → (((𝐴 ↾ 𝑋)‘𝑦) ≠ ((𝐵 ↾ 𝑋)‘𝑦) → 𝑦 ∈ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}))
68	67	con3d 147	. . . . . . . 8 ⊢ (𝑦 ∈ On → (¬ 𝑦 ∈ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} → ¬ ((𝐴 ↾ 𝑋)‘𝑦) ≠ ((𝐵 ↾ 𝑋)‘𝑦)))
69	55, 62, 68	sylc 63	. . . . . . 7 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ (𝐴 ↾ 𝑋) <s (𝐵 ↾ 𝑋)) ∧ 𝑦 ∈ ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) → ¬ ((𝐴 ↾ 𝑋)‘𝑦) ≠ ((𝐵 ↾ 𝑋)‘𝑦))
70		df-ne 2782	. . . . . . . 8 ⊢ (((𝐴 ↾ 𝑋)‘𝑦) ≠ ((𝐵 ↾ 𝑋)‘𝑦) ↔ ¬ ((𝐴 ↾ 𝑋)‘𝑦) = ((𝐵 ↾ 𝑋)‘𝑦))
71	70	con2bii 346	. . . . . . 7 ⊢ (((𝐴 ↾ 𝑋)‘𝑦) = ((𝐵 ↾ 𝑋)‘𝑦) ↔ ¬ ((𝐴 ↾ 𝑋)‘𝑦) ≠ ((𝐵 ↾ 𝑋)‘𝑦))
72	69, 71	sylibr 223	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ (𝐴 ↾ 𝑋) <s (𝐵 ↾ 𝑋)) ∧ 𝑦 ∈ ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) → ((𝐴 ↾ 𝑋)‘𝑦) = ((𝐵 ↾ 𝑋)‘𝑦))
73		fvres 6117	. . . . . . . 8 ⊢ (𝑦 ∈ 𝑋 → ((𝐴 ↾ 𝑋)‘𝑦) = (𝐴‘𝑦))
74		fvres 6117	. . . . . . . 8 ⊢ (𝑦 ∈ 𝑋 → ((𝐵 ↾ 𝑋)‘𝑦) = (𝐵‘𝑦))
75	73, 74	eqeq12d 2625	. . . . . . 7 ⊢ (𝑦 ∈ 𝑋 → (((𝐴 ↾ 𝑋)‘𝑦) = ((𝐵 ↾ 𝑋)‘𝑦) ↔ (𝐴‘𝑦) = (𝐵‘𝑦)))
76	75	biimpd 218	. . . . . 6 ⊢ (𝑦 ∈ 𝑋 → (((𝐴 ↾ 𝑋)‘𝑦) = ((𝐵 ↾ 𝑋)‘𝑦) → (𝐴‘𝑦) = (𝐵‘𝑦)))
77	53, 72, 76	sylc 63	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ (𝐴 ↾ 𝑋) <s (𝐵 ↾ 𝑋)) ∧ 𝑦 ∈ ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) → (𝐴‘𝑦) = (𝐵‘𝑦))
78	77	ralrimiva 2949	. . . 4 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ (𝐴 ↾ 𝑋) <s (𝐵 ↾ 𝑋)) → ∀𝑦 ∈ ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} (𝐴‘𝑦) = (𝐵‘𝑦))
79		fvresval 30911	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = (𝐴‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) ∨ ((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅)
80	79	ori 389	. . . . . . . . . . . . . 14 ⊢ (¬ ((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = (𝐴‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) → ((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅)
81	19, 80	nsyl2 141	. . . . . . . . . . . . 13 ⊢ (((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 1𝑜 → ((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = (𝐴‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}))
82	81	eqcomd 2616	. . . . . . . . . . . 12 ⊢ (((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 1𝑜 → (𝐴‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}))
83		eqeq2 2621	. . . . . . . . . . . 12 ⊢ (((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 1𝑜 → ((𝐴‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) ↔ (𝐴‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 1𝑜))
84	82, 83	mpbid 221	. . . . . . . . . . 11 ⊢ (((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 1𝑜 → (𝐴‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 1𝑜)
85	84	adantr 480	. . . . . . . . . 10 ⊢ ((((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅) → (𝐴‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 1𝑜)
86	85	a1i 11	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) → ((((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅) → (𝐴‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 1𝑜))
87	21	ad2antrl 760	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ (((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅)) → ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom (𝐴 ↾ 𝑋))
88	87, 45	syl 17	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ (((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅)) → ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ 𝑋)
89		nofun 31046	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵 ↾ 𝑋) ∈ No → Fun (𝐵 ↾ 𝑋))
90		fvelrn 6260	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Fun (𝐵 ↾ 𝑋) ∧ ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom (𝐵 ↾ 𝑋)) → ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) ∈ ran (𝐵 ↾ 𝑋))
91	90	ex 449	. . . . . . . . . . . . . . . . . 18 ⊢ (Fun (𝐵 ↾ 𝑋) → (∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom (𝐵 ↾ 𝑋) → ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) ∈ ran (𝐵 ↾ 𝑋)))
92	89, 91	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐵 ↾ 𝑋) ∈ No → (∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom (𝐵 ↾ 𝑋) → ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) ∈ ran (𝐵 ↾ 𝑋)))
93		norn 31048	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵 ↾ 𝑋) ∈ No → ran (𝐵 ↾ 𝑋) ⊆ {1𝑜, 2𝑜})
94	93	sseld 3567	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐵 ↾ 𝑋) ∈ No → (((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) ∈ ran (𝐵 ↾ 𝑋) → ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) ∈ {1𝑜, 2𝑜}))
95	92, 94	syld 46	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ↾ 𝑋) ∈ No → (∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom (𝐵 ↾ 𝑋) → ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) ∈ {1𝑜, 2𝑜}))
96		nosgnn0 31055	. . . . . . . . . . . . . . . . 17 ⊢ ¬ ∅ ∈ {1𝑜, 2𝑜}
97		eleq1 2676	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅ → (((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) ∈ {1𝑜, 2𝑜} ↔ ∅ ∈ {1𝑜, 2𝑜}))
98	96, 97	mtbiri 316	. . . . . . . . . . . . . . . 16 ⊢ (((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅ → ¬ ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) ∈ {1𝑜, 2𝑜})
99	95, 98	nsyli 154	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ↾ 𝑋) ∈ No → (((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅ → ¬ ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom (𝐵 ↾ 𝑋)))
100	4, 99	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) → (((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅ → ¬ ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom (𝐵 ↾ 𝑋)))
101	100	imp 444	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅) → ¬ ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom (𝐵 ↾ 𝑋))
102	101	adantrl 748	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ (((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅)) → ¬ ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom (𝐵 ↾ 𝑋))
103	47	simplbi2 653	. . . . . . . . . . . . 13 ⊢ (∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ 𝑋 → (∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom 𝐵 → ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom (𝐵 ↾ 𝑋)))
104	103	con3d 147	. . . . . . . . . . . 12 ⊢ (∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ 𝑋 → (¬ ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom (𝐵 ↾ 𝑋) → ¬ ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom 𝐵))
105	88, 102, 104	sylc 63	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ (((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅)) → ¬ ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom 𝐵)
106		ndmfv 6128	. . . . . . . . . . 11 ⊢ (¬ ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom 𝐵 → (𝐵‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅)
107	105, 106	syl 17	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ (((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅)) → (𝐵‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅)
108	107	ex 449	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) → ((((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅) → (𝐵‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅))
109	86, 108	jcad 554	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) → ((((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅) → ((𝐴‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 1𝑜 ∧ (𝐵‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅)))
110		fvresval 30911	. . . . . . . . . . . . . 14 ⊢ (((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = (𝐵‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) ∨ ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅)
111	110	ori 389	. . . . . . . . . . . . 13 ⊢ (¬ ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = (𝐵‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) → ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅)
112	34, 111	nsyl2 141	. . . . . . . . . . . 12 ⊢ (((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 2𝑜 → ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = (𝐵‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}))
113	112	eqcomd 2616	. . . . . . . . . . 11 ⊢ (((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 2𝑜 → (𝐵‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}))
114		eqeq2 2621	. . . . . . . . . . 11 ⊢ (((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 2𝑜 → ((𝐵‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) ↔ (𝐵‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 2𝑜))
115	113, 114	mpbid 221	. . . . . . . . . 10 ⊢ (((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 2𝑜 → (𝐵‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 2𝑜)
116	84, 115	anim12i 588	. . . . . . . . 9 ⊢ ((((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 2𝑜) → ((𝐴‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 1𝑜 ∧ (𝐵‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 2𝑜))
117	116	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) → ((((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 2𝑜) → ((𝐴‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 1𝑜 ∧ (𝐵‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 2𝑜)))
118	36	ad2antll 761	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ (((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅ ∧ ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 2𝑜)) → ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom (𝐵 ↾ 𝑋))
119	118, 48	syl 17	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ (((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅ ∧ ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 2𝑜)) → ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ 𝑋)
120		nofun 31046	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ↾ 𝑋) ∈ No → Fun (𝐴 ↾ 𝑋))
121		fvelrn 6260	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Fun (𝐴 ↾ 𝑋) ∧ ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom (𝐴 ↾ 𝑋)) → ((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) ∈ ran (𝐴 ↾ 𝑋))
122	121	ex 449	. . . . . . . . . . . . . . . . . 18 ⊢ (Fun (𝐴 ↾ 𝑋) → (∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom (𝐴 ↾ 𝑋) → ((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) ∈ ran (𝐴 ↾ 𝑋)))
123	120, 122	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ↾ 𝑋) ∈ No → (∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom (𝐴 ↾ 𝑋) → ((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) ∈ ran (𝐴 ↾ 𝑋)))
124		norn 31048	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ↾ 𝑋) ∈ No → ran (𝐴 ↾ 𝑋) ⊆ {1𝑜, 2𝑜})
125	124	sseld 3567	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ↾ 𝑋) ∈ No → (((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) ∈ ran (𝐴 ↾ 𝑋) → ((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) ∈ {1𝑜, 2𝑜}))
126	123, 125	syld 46	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ↾ 𝑋) ∈ No → (∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom (𝐴 ↾ 𝑋) → ((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) ∈ {1𝑜, 2𝑜}))
127		eleq1 2676	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅ → (((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) ∈ {1𝑜, 2𝑜} ↔ ∅ ∈ {1𝑜, 2𝑜}))
128	96, 127	mtbiri 316	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅ → ¬ ((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) ∈ {1𝑜, 2𝑜})
129	126, 128	nsyli 154	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ↾ 𝑋) ∈ No → (((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅ → ¬ ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom (𝐴 ↾ 𝑋)))
130	2, 129	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) → (((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅ → ¬ ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom (𝐴 ↾ 𝑋)))
131	130	imp 444	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅) → ¬ ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom (𝐴 ↾ 𝑋))
132	131	adantrr 749	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ (((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅ ∧ ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 2𝑜)) → ¬ ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom (𝐴 ↾ 𝑋))
133	44	simplbi2 653	. . . . . . . . . . . . 13 ⊢ (∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ 𝑋 → (∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom 𝐴 → ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom (𝐴 ↾ 𝑋)))
134	133	con3d 147	. . . . . . . . . . . 12 ⊢ (∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ 𝑋 → (¬ ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom (𝐴 ↾ 𝑋) → ¬ ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom 𝐴))
135	119, 132, 134	sylc 63	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ (((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅ ∧ ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 2𝑜)) → ¬ ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom 𝐴)
136	135	ex 449	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) → ((((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅ ∧ ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 2𝑜) → ¬ ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom 𝐴))
137		ndmfv 6128	. . . . . . . . . 10 ⊢ (¬ ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom 𝐴 → (𝐴‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅)
138	136, 137	syl6 34	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) → ((((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅ ∧ ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 2𝑜) → (𝐴‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅))
139	115	adantl 481	. . . . . . . . . 10 ⊢ ((((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅ ∧ ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 2𝑜) → (𝐵‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 2𝑜)
140	139	a1i 11	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) → ((((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅ ∧ ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 2𝑜) → (𝐵‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 2𝑜))
141	138, 140	jcad 554	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) → ((((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅ ∧ ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 2𝑜) → ((𝐴‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅ ∧ (𝐵‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 2𝑜)))
142	109, 117, 141	3orim123d 1399	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) → (((((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅) ∨ (((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 1𝑜 ∧ ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 2𝑜) ∨ (((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅ ∧ ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 2𝑜)) → (((𝐴‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 1𝑜 ∧ (𝐵‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅) ∨ ((𝐴‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 1𝑜 ∧ (𝐵‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 2𝑜) ∨ ((𝐴‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅ ∧ (𝐵‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 2𝑜))))
143		fvex 6113	. . . . . . . 8 ⊢ (𝐴‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) ∈ V
144		fvex 6113	. . . . . . . 8 ⊢ (𝐵‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) ∈ V
145	143, 144	brtp 30892	. . . . . . 7 ⊢ ((𝐴‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) ↔ (((𝐴‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 1𝑜 ∧ (𝐵‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅) ∨ ((𝐴‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 1𝑜 ∧ (𝐵‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 2𝑜) ∨ ((𝐴‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅ ∧ (𝐵‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 2𝑜)))
146	142, 15, 145	3imtr4g 284	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) → (((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) → (𝐴‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)})))
147	12, 146	sylbid 229	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) → ((𝐴 ↾ 𝑋) <s (𝐵 ↾ 𝑋) → (𝐴‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)})))
148	147	imp 444	. . . 4 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ (𝐴 ↾ 𝑋) <s (𝐵 ↾ 𝑋)) → (𝐴‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}))
149		raleq 3115	. . . . . 6 ⊢ (𝑥 = ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} → (∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦) ↔ ∀𝑦 ∈ ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} (𝐴‘𝑦) = (𝐵‘𝑦)))
150		fveq2 6103	. . . . . . 7 ⊢ (𝑥 = ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} → (𝐴‘𝑥) = (𝐴‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}))
151		fveq2 6103	. . . . . . 7 ⊢ (𝑥 = ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} → (𝐵‘𝑥) = (𝐵‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}))
152	150, 151	breq12d 4596	. . . . . 6 ⊢ (𝑥 = ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} → ((𝐴‘𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵‘𝑥) ↔ (𝐴‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)})))
153	149, 152	anbi12d 743	. . . . 5 ⊢ (𝑥 = ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} → ((∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦) ∧ (𝐴‘𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵‘𝑥)) ↔ (∀𝑦 ∈ ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} (𝐴‘𝑦) = (𝐵‘𝑦) ∧ (𝐴‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}))))
154	153	rspcev 3282	. . . 4 ⊢ ((∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ On ∧ (∀𝑦 ∈ ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} (𝐴‘𝑦) = (𝐵‘𝑦) ∧ (𝐴‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}))) → ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦) ∧ (𝐴‘𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵‘𝑥)))
155	9, 78, 148, 154	syl12anc 1316	. . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ (𝐴 ↾ 𝑋) <s (𝐵 ↾ 𝑋)) → ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦) ∧ (𝐴‘𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵‘𝑥)))
156		sltval 31044	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 <s 𝐵 ↔ ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦) ∧ (𝐴‘𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵‘𝑥))))
157	156	3adant3 1074	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) → (𝐴 <s 𝐵 ↔ ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦) ∧ (𝐴‘𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵‘𝑥))))
158	157	adantr 480	. . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ (𝐴 ↾ 𝑋) <s (𝐵 ↾ 𝑋)) → (𝐴 <s 𝐵 ↔ ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦) ∧ (𝐴‘𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐵‘𝑥))))
159	155, 158	mpbird 246	. 2 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ (𝐴 ↾ 𝑋) <s (𝐵 ↾ 𝑋)) → 𝐴 <s 𝐵)
160	159	ex 449	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) → ((𝐴 ↾ 𝑋) <s (𝐵 ↾ 𝑋) → 𝐴 <s 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∨ w3o 1030   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  {cpr 4127  {ctp 4129  ⟨cop 4131  ∩ cint 4410   class class class wbr 4583  dom cdm 5038  ran crn 5039   ↾ cres 5040  Oncon0 5640  Fun wfun 5798  ‘cfv 5804  1_𝑜c1o 7440  2_𝑜c2o 7441   No csur 31037   <s cslt 31038

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-1o 7447  df-2o 7448  df-no 31040  df-slt 31041

This theorem is referenced by: (None)