Theorem sltres 33164

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 33162	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑋 ∈ On) → (𝐴 ↾ 𝑋) ∈ No )
2	1	3adant2 1127	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) → (𝐴 ↾ 𝑋) ∈ No )
3		noreson 33162	. . . . . . 7 ⊢ ((𝐵 ∈ No ∧ 𝑋 ∈ On) → (𝐵 ↾ 𝑋) ∈ No )
4	3	3adant1 1126	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) → (𝐵 ↾ 𝑋) ∈ No )
5		sltintdifex 33163	. . . . . . 7 ⊢ (((𝐴 ↾ 𝑋) ∈ No ∧ (𝐵 ↾ 𝑋) ∈ No ) → ((𝐴 ↾ 𝑋) <s (𝐵 ↾ 𝑋) → ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ V))
6		onintrab 7510	. . . . . . 7 ⊢ (∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ V ↔ ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ On)
7	5, 6	syl6ib 253	. . . . . 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 586	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) → ((𝐴 ↾ 𝑋) <s (𝐵 ↾ 𝑋) → ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ On))
9	8	imp 409	. . . 4 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ (𝐴 ↾ 𝑋) <s (𝐵 ↾ 𝑋)) → ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ On)
10		simpl3 1189	. . . . . . . 8 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ (𝐴 ↾ 𝑋) <s (𝐵 ↾ 𝑋)) → 𝑋 ∈ On)
11		sltval2 33158	. . . . . . . . . . . 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 586	. . . . . . . . . . 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 6678	. . . . . . . . . . . . 13 ⊢ ((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) ∈ V
14		fvex 6678	. . . . . . . . . . . . 13 ⊢ ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) ∈ V
15	13, 14	brtp 32980	. . . . . . . . . . . 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 8113	. . . . . . . . . . . . . . . . . 18 ⊢ 1o ≠ ∅
17	16	neii 3018	. . . . . . . . . . . . . . . . 17 ⊢ ¬ 1o = ∅
18		eqeq1 2825	. . . . . . . . . . . . . . . . 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 329	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 1o → ¬ ((𝐴 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅)
20		ndmfv 6695	. . . . . . . . . . . . . . . 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 143	. . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . 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 869	. . . . . . . . . . . . 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 483	. . . . . . . . . . . . . 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 869	. . . . . . . . . . . . 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 8105	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 2o ∈ On
27	26	elexi 3514	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2o ∈ V
28	27	prid2 4693	. . . . . . . . . . . . . . . . . . 19 ⊢ 2o ∈ {1o, 2o}
29	28	nosgnn0i 33161	. . . . . . . . . . . . . . . . . 18 ⊢ ∅ ≠ 2o
30	29	neii 3018	. . . . . . . . . . . . . . . . 17 ⊢ ¬ ∅ = 2o
31		eqeq1 2825	. . . . . . . . . . . . . . . . . 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 2828	. . . . . . . . . . . . . . . . . 18 ⊢ (2o = ∅ ↔ ∅ = 2o)
33	31, 32	syl6bb 289	. . . . . . . . . . . . . . . . 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 329	. . . . . . . . . . . . . . . 16 ⊢ (((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = 2o → ¬ ((𝐵 ↾ 𝑋)‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅)
35		ndmfv 6695	. . . . . . . . . . . . . . . 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 143	. . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . 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 870	. . . . . . . . . . . . 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 1423	. . . . . . . . . . . 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 219	. . . . . . . . . . 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 255	. . . . . . . . . 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 409	. . . . . . . . 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 5870	. . . . . . . . . . . 12 ⊢ dom (𝐴 ↾ 𝑋) = (𝑋 ∩ dom 𝐴)
44	43	elin2 4174	. . . . . . . . . . 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 500	. . . . . . . . . 10 ⊢ (∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom (𝐴 ↾ 𝑋) → ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ 𝑋)
46		dmres 5870	. . . . . . . . . . . 12 ⊢ dom (𝐵 ↾ 𝑋) = (𝑋 ∩ dom 𝐵)
47	46	elin2 4174	. . . . . . . . . . 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 500	. . . . . . . . . 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 853	. . . . . . . . 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 6228	. . . . . . . 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 65	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ (𝐴 ↾ 𝑋) <s (𝐵 ↾ 𝑋)) → ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ⊆ 𝑋)
53	52	sselda 3967	. . . . . 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 6211	. . . . . . . . 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 582	. . . . . . . 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 4884	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} → ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ⊆ 𝑦)
57		ontri1 6220	. . . . . . . . . . . . 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 246	. . . . . . . . . . . 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 136	. . . . . . . . . . 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 582	. . . . . . . . . 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 454	. . . . . . . . 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 6665	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑦 → ((𝐴 ↾ 𝑋)‘𝑎) = ((𝐴 ↾ 𝑋)‘𝑦))
64		fveq2 6665	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑦 → ((𝐵 ↾ 𝑋)‘𝑎) = ((𝐵 ↾ 𝑋)‘𝑦))
65	63, 64	neeq12d 3077	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑦 → (((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎) ↔ ((𝐴 ↾ 𝑋)‘𝑦) ≠ ((𝐵 ↾ 𝑋)‘𝑦)))
66	65	elrab 3680	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ↔ (𝑦 ∈ On ∧ ((𝐴 ↾ 𝑋)‘𝑦) ≠ ((𝐵 ↾ 𝑋)‘𝑦)))
67	66	simplbi2 503	. . . . . . . . 9 ⊢ (𝑦 ∈ On → (((𝐴 ↾ 𝑋)‘𝑦) ≠ ((𝐵 ↾ 𝑋)‘𝑦) → 𝑦 ∈ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}))
68	67	con3d 155	. . . . . . . 8 ⊢ (𝑦 ∈ On → (¬ 𝑦 ∈ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} → ¬ ((𝐴 ↾ 𝑋)‘𝑦) ≠ ((𝐵 ↾ 𝑋)‘𝑦)))
69	55, 62, 68	sylc 65	. . . . . . 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 3017	. . . . . . . 8 ⊢ (((𝐴 ↾ 𝑋)‘𝑦) ≠ ((𝐵 ↾ 𝑋)‘𝑦) ↔ ¬ ((𝐴 ↾ 𝑋)‘𝑦) = ((𝐵 ↾ 𝑋)‘𝑦))
71	70	con2bii 360	. . . . . . 7 ⊢ (((𝐴 ↾ 𝑋)‘𝑦) = ((𝐵 ↾ 𝑋)‘𝑦) ↔ ¬ ((𝐴 ↾ 𝑋)‘𝑦) ≠ ((𝐵 ↾ 𝑋)‘𝑦))
72	69, 71	sylibr 236	. . . . . 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 6684	. . . . . . . 8 ⊢ (𝑦 ∈ 𝑋 → ((𝐴 ↾ 𝑋)‘𝑦) = (𝐴‘𝑦))
74		fvres 6684	. . . . . . . 8 ⊢ (𝑦 ∈ 𝑋 → ((𝐵 ↾ 𝑋)‘𝑦) = (𝐵‘𝑦))
75	73, 74	eqeq12d 2837	. . . . . . 7 ⊢ (𝑦 ∈ 𝑋 → (((𝐴 ↾ 𝑋)‘𝑦) = ((𝐵 ↾ 𝑋)‘𝑦) ↔ (𝐴‘𝑦) = (𝐵‘𝑦)))
76	75	biimpd 231	. . . . . 6 ⊢ (𝑦 ∈ 𝑋 → (((𝐴 ↾ 𝑋)‘𝑦) = ((𝐵 ↾ 𝑋)‘𝑦) → (𝐴‘𝑦) = (𝐵‘𝑦)))
77	53, 72, 76	sylc 65	. . . . 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 3182	. . . 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 33005	. . . . . . . . . . . . . . 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 857	. . . . . . . . . . . . . 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 143	. . . . . . . . . . . . 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 2827	. . . . . . . . . . . 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 2833	. . . . . . . . . . . 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 234	. . . . . . . . . . 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 483	. . . . . . . . . 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 726	. . . . . . . . . . . . 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 33151	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵 ↾ 𝑋) ∈ No → Fun (𝐵 ↾ 𝑋))
90		fvelrn 6839	. . . . . . . . . . . . . . . . . . 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 415	. . . . . . . . . . . . . . . . . 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 33153	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵 ↾ 𝑋) ∈ No → ran (𝐵 ↾ 𝑋) ⊆ {1o, 2o})
94	93	sseld 3966	. . . . . . . . . . . . . . . . 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 47	. . . . . . . . . . . . . . . 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 33160	. . . . . . . . . . . . . . . . 17 ⊢ ¬ ∅ ∈ {1o, 2o}
97		eleq1 2900	. . . . . . . . . . . . . . . . 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 329	. . . . . . . . . . . . . . . 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 160	. . . . . . . . . . . . . . 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 409	. . . . . . . . . . . . 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 714	. . . . . . . . . . . 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 503	. . . . . . . . . . . . 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 155	. . . . . . . . . . . 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 65	. . . . . . . . . . 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 6695	. . . . . . . . . . 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 415	. . . . . . . . 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 515	. . . . . . . 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 33005	. . . . . . . . . . . . . 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 857	. . . . . . . . . . . . 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 143	. . . . . . . . . . . 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 2827	. . . . . . . . . . 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 2833	. . . . . . . . . . 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 234	. . . . . . . . . 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 614	. . . . . . . . 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 727	. . . . . . . . . . . . 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 33151	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ↾ 𝑋) ∈ No → Fun (𝐴 ↾ 𝑋))
121		fvelrn 6839	. . . . . . . . . . . . . . . . . . 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 415	. . . . . . . . . . . . . . . . . 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 33153	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ↾ 𝑋) ∈ No → ran (𝐴 ↾ 𝑋) ⊆ {1o, 2o})
125	124	sseld 3966	. . . . . . . . . . . . . . . . 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 47	. . . . . . . . . . . . . . . 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 2900	. . . . . . . . . . . . . . . . 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 329	. . . . . . . . . . . . . . . 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 160	. . . . . . . . . . . . . . 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 409	. . . . . . . . . . . . 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 715	. . . . . . . . . . . 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 503	. . . . . . . . . . . . 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 155	. . . . . . . . . . . 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 65	. . . . . . . . . . 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 415	. . . . . . . . . 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 6695	. . . . . . . . . 10 ⊢ (¬ ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} ∈ dom 𝐴 → (𝐴‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) = ∅)
138	136, 137	syl6 35	. . . . . . . . 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 484	. . . . . . . . . 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 515	. . . . . . . 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 1440	. . . . . . 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 6678	. . . . . . . 8 ⊢ (𝐴‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) ∈ V
144		fvex 6678	. . . . . . . 8 ⊢ (𝐵‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}) ∈ V
145	143, 144	brtp 32980	. . . . . . 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 298	. . . . . 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 242	. . . . 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 409	. . . 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 3406	. . . . . 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 6665	. . . . . . 7 ⊢ (𝑥 = ∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)} → (𝐴‘𝑥) = (𝐴‘∩ {𝑎 ∈ On ∣ ((𝐴 ↾ 𝑋)‘𝑎) ≠ ((𝐵 ↾ 𝑋)‘𝑎)}))
151		fveq2 6665	. . . . . . 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 5072	. . . . . 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 632	. . . . 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 3623	. . . 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 834	. . 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 33149	. . . . 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 1128	. . . 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 483	. . 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 259	. 2 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ (𝐴 ↾ 𝑋) <s (𝐵 ↾ 𝑋)) → 𝐴 <s 𝐵)
160	159	ex 415	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) → ((𝐴 ↾ 𝑋) <s (𝐵 ↾ 𝑋) → 𝐴 <s 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   ∨ w3o 1082   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110   ≠ wne 3016  ∀wral 3138  ∃wrex 3139  {crab 3142  Vcvv 3495   ⊆ wss 3936  ∅c0 4291  {cpr 4563  {ctp 4565  ⟨cop 4567  ∩ cint 4869   class class class wbr 5059  dom cdm 5550  ran crn 5551   ↾ cres 5552  Oncon0 6186  Fun wfun 6344  ‘cfv 6350  1_oc1o 8089  2_oc2o 8090   No csur 33142   <s cslt 33143

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4833  df-int 4870  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5455  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-we 5511  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-ord 6189  df-on 6190  df-suc 6192  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-1o 8096  df-2o 8097  df-no 33145  df-slt 33146

This theorem is referenced by:  noresle  33195  nosupbnd1lem1  33203  nosupbnd1lem2  33204  nosupbnd1  33209  nosupbnd2lem1  33210  nosupbnd2  33211  noetalem2  33213  noetalem3  33214