Theorem nolt02o 27198

Description: Given 𝐴 less-than 𝐵, equal to 𝐵 up to 𝑋, and undefined at 𝑋, then 𝐵(𝑋) = 2_o. (Contributed by Scott Fenton, 6-Dec-2021.)

Assertion

Ref	Expression
nolt02o	⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) → (𝐵‘𝑋) = 2o)

Proof of Theorem nolt02o

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

Step	Hyp	Ref	Expression
1		simp11 1204	. . . . 5 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) → 𝐴 ∈ No )
2		sltso 27179	. . . . . 6 ⊢ <s Or No
3		sonr 5612	. . . . . 6 ⊢ (( <s Or No ∧ 𝐴 ∈ No ) → ¬ 𝐴 <s 𝐴)
4	2, 3	mpan 689	. . . . 5 ⊢ (𝐴 ∈ No → ¬ 𝐴 <s 𝐴)
5	1, 4	syl 17	. . . 4 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) → ¬ 𝐴 <s 𝐴)
6		simp2r 1201	. . . . 5 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) → 𝐴 <s 𝐵)
7		breq2 5153	. . . . 5 ⊢ (𝐴 = 𝐵 → (𝐴 <s 𝐴 ↔ 𝐴 <s 𝐵))
8	6, 7	syl5ibrcom 246	. . . 4 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) → (𝐴 = 𝐵 → 𝐴 <s 𝐴))
9	5, 8	mtod 197	. . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) → ¬ 𝐴 = 𝐵)
10		simpl2l 1227	. . . 4 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) ∧ (𝐵‘𝑋) = ∅) → (𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋))
11		simpl11 1249	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) ∧ (𝐵‘𝑋) = ∅) → 𝐴 ∈ No )
12		nofun 27152	. . . . . 6 ⊢ (𝐴 ∈ No → Fun 𝐴)
13		funrel 6566	. . . . . 6 ⊢ (Fun 𝐴 → Rel 𝐴)
14	11, 12, 13	3syl 18	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) ∧ (𝐵‘𝑋) = ∅) → Rel 𝐴)
15		simpl13 1251	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) ∧ (𝐵‘𝑋) = ∅) → 𝑋 ∈ On)
16		simpl3 1194	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) ∧ (𝐵‘𝑋) = ∅) → (𝐴‘𝑋) = ∅)
17		nolt02olem 27197	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑋 ∈ On ∧ (𝐴‘𝑋) = ∅) → dom 𝐴 ⊆ 𝑋)
18	11, 15, 16, 17	syl3anc 1372	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) ∧ (𝐵‘𝑋) = ∅) → dom 𝐴 ⊆ 𝑋)
19		relssres 6023	. . . . 5 ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ 𝑋) → (𝐴 ↾ 𝑋) = 𝐴)
20	14, 18, 19	syl2anc 585	. . . 4 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) ∧ (𝐵‘𝑋) = ∅) → (𝐴 ↾ 𝑋) = 𝐴)
21		simpl12 1250	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) ∧ (𝐵‘𝑋) = ∅) → 𝐵 ∈ No )
22		nofun 27152	. . . . . 6 ⊢ (𝐵 ∈ No → Fun 𝐵)
23		funrel 6566	. . . . . 6 ⊢ (Fun 𝐵 → Rel 𝐵)
24	21, 22, 23	3syl 18	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) ∧ (𝐵‘𝑋) = ∅) → Rel 𝐵)
25		simpr 486	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) ∧ (𝐵‘𝑋) = ∅) → (𝐵‘𝑋) = ∅)
26		nolt02olem 27197	. . . . . 6 ⊢ ((𝐵 ∈ No ∧ 𝑋 ∈ On ∧ (𝐵‘𝑋) = ∅) → dom 𝐵 ⊆ 𝑋)
27	21, 15, 25, 26	syl3anc 1372	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) ∧ (𝐵‘𝑋) = ∅) → dom 𝐵 ⊆ 𝑋)
28		relssres 6023	. . . . 5 ⊢ ((Rel 𝐵 ∧ dom 𝐵 ⊆ 𝑋) → (𝐵 ↾ 𝑋) = 𝐵)
29	24, 27, 28	syl2anc 585	. . . 4 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) ∧ (𝐵‘𝑋) = ∅) → (𝐵 ↾ 𝑋) = 𝐵)
30	10, 20, 29	3eqtr3d 2781	. . 3 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) ∧ (𝐵‘𝑋) = ∅) → 𝐴 = 𝐵)
31	9, 30	mtand 815	. 2 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) → ¬ (𝐵‘𝑋) = ∅)
32		simp12 1205	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) → 𝐵 ∈ No )
33		sltval 27150	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 <s 𝐵 ↔ ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦) ∧ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑥))))
34	1, 32, 33	syl2anc 585	. . . . 5 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) → (𝐴 <s 𝐵 ↔ ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦) ∧ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑥))))
35	6, 34	mpbid 231	. . . 4 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) → ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦) ∧ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑥)))
36		df-an 398	. . . . . 6 ⊢ ((∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦) ∧ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑥)) ↔ ¬ (∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦) → ¬ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑥)))
37	36	rexbii 3095	. . . . 5 ⊢ (∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦) ∧ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑥)) ↔ ∃𝑥 ∈ On ¬ (∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦) → ¬ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑥)))
38		rexnal 3101	. . . . 5 ⊢ (∃𝑥 ∈ On ¬ (∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦) → ¬ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑥)) ↔ ¬ ∀𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦) → ¬ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑥)))
39	37, 38	bitri 275	. . . 4 ⊢ (∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦) ∧ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑥)) ↔ ¬ ∀𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦) → ¬ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑥)))
40	35, 39	sylib 217	. . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) → ¬ ∀𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦) → ¬ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑥)))
41		1oex 8476	. . . . . . . . . . . 12 ⊢ 1o ∈ V
42	41	prid1 4767	. . . . . . . . . . 11 ⊢ 1o ∈ {1o, 2o}
43	42	nosgnn0i 27162	. . . . . . . . . 10 ⊢ ∅ ≠ 1o
44	43	neii 2943	. . . . . . . . 9 ⊢ ¬ ∅ = 1o
45		simpll3 1215	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) ∧ (𝐵‘𝑋) = 1o) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦))) → (𝐴‘𝑋) = ∅)
46		simplr 768	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) ∧ (𝐵‘𝑋) = 1o) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦))) → (𝐵‘𝑋) = 1o)
47		eqeq1 2737	. . . . . . . . . . . . 13 ⊢ ((𝐴‘𝑋) = (𝐵‘𝑋) → ((𝐴‘𝑋) = ∅ ↔ (𝐵‘𝑋) = ∅))
48	47	anbi1d 631	. . . . . . . . . . . 12 ⊢ ((𝐴‘𝑋) = (𝐵‘𝑋) → (((𝐴‘𝑋) = ∅ ∧ (𝐵‘𝑋) = 1o) ↔ ((𝐵‘𝑋) = ∅ ∧ (𝐵‘𝑋) = 1o)))
49		eqtr2 2757	. . . . . . . . . . . 12 ⊢ (((𝐵‘𝑋) = ∅ ∧ (𝐵‘𝑋) = 1o) → ∅ = 1o)
50	48, 49	syl6bi 253	. . . . . . . . . . 11 ⊢ ((𝐴‘𝑋) = (𝐵‘𝑋) → (((𝐴‘𝑋) = ∅ ∧ (𝐵‘𝑋) = 1o) → ∅ = 1o))
51	50	com12 32	. . . . . . . . . 10 ⊢ (((𝐴‘𝑋) = ∅ ∧ (𝐵‘𝑋) = 1o) → ((𝐴‘𝑋) = (𝐵‘𝑋) → ∅ = 1o))
52	45, 46, 51	syl2anc 585	. . . . . . . . 9 ⊢ (((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) ∧ (𝐵‘𝑋) = 1o) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦))) → ((𝐴‘𝑋) = (𝐵‘𝑋) → ∅ = 1o))
53	44, 52	mtoi 198	. . . . . . . 8 ⊢ (((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) ∧ (𝐵‘𝑋) = 1o) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦))) → ¬ (𝐴‘𝑋) = (𝐵‘𝑋))
54		simpr 486	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) ∧ (𝐵‘𝑋) = 1o) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦))) ∧ 𝑋 ∈ 𝑥) → 𝑋 ∈ 𝑥)
55		simplrr 777	. . . . . . . . 9 ⊢ ((((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) ∧ (𝐵‘𝑋) = 1o) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦))) ∧ 𝑋 ∈ 𝑥) → ∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦))
56		fveq2 6892	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑋 → (𝐴‘𝑦) = (𝐴‘𝑋))
57		fveq2 6892	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑋 → (𝐵‘𝑦) = (𝐵‘𝑋))
58	56, 57	eqeq12d 2749	. . . . . . . . . 10 ⊢ (𝑦 = 𝑋 → ((𝐴‘𝑦) = (𝐵‘𝑦) ↔ (𝐴‘𝑋) = (𝐵‘𝑋)))
59	58	rspcv 3609	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝑥 → (∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦) → (𝐴‘𝑋) = (𝐵‘𝑋)))
60	54, 55, 59	sylc 65	. . . . . . . 8 ⊢ ((((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) ∧ (𝐵‘𝑋) = 1o) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦))) ∧ 𝑋 ∈ 𝑥) → (𝐴‘𝑋) = (𝐵‘𝑋))
61	53, 60	mtand 815	. . . . . . 7 ⊢ (((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) ∧ (𝐵‘𝑋) = 1o) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦))) → ¬ 𝑋 ∈ 𝑥)
62		simprl 770	. . . . . . . 8 ⊢ (((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) ∧ (𝐵‘𝑋) = 1o) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦))) → 𝑥 ∈ On)
63		simpl13 1251	. . . . . . . . 9 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) ∧ (𝐵‘𝑋) = 1o) → 𝑋 ∈ On)
64	63	adantr 482	. . . . . . . 8 ⊢ (((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) ∧ (𝐵‘𝑋) = 1o) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦))) → 𝑋 ∈ On)
65		ontri1 6399	. . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ 𝑋 ∈ On) → (𝑥 ⊆ 𝑋 ↔ ¬ 𝑋 ∈ 𝑥))
66	62, 64, 65	syl2anc 585	. . . . . . 7 ⊢ (((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) ∧ (𝐵‘𝑋) = 1o) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦))) → (𝑥 ⊆ 𝑋 ↔ ¬ 𝑋 ∈ 𝑥))
67	61, 66	mpbird 257	. . . . . 6 ⊢ (((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) ∧ (𝐵‘𝑋) = 1o) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦))) → 𝑥 ⊆ 𝑋)
68		onsseleq 6406	. . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ 𝑋 ∈ On) → (𝑥 ⊆ 𝑋 ↔ (𝑥 ∈ 𝑋 ∨ 𝑥 = 𝑋)))
69	62, 64, 68	syl2anc 585	. . . . . . 7 ⊢ (((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) ∧ (𝐵‘𝑋) = 1o) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦))) → (𝑥 ⊆ 𝑋 ↔ (𝑥 ∈ 𝑋 ∨ 𝑥 = 𝑋)))
70		eqtr2 2757	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ↾ 𝑋)‘𝑥) = ∅ ∧ ((𝐴 ↾ 𝑋)‘𝑥) = 1o) → ∅ = 1o)
71	70	ancoms 460	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ↾ 𝑋)‘𝑥) = 1o ∧ ((𝐴 ↾ 𝑋)‘𝑥) = ∅) → ∅ = 1o)
72	44, 71	mto 196	. . . . . . . . . . . 12 ⊢ ¬ (((𝐴 ↾ 𝑋)‘𝑥) = 1o ∧ ((𝐴 ↾ 𝑋)‘𝑥) = ∅)
73		df-1o 8466	. . . . . . . . . . . . . . . 16 ⊢ 1o = suc ∅
74		df-2o 8467	. . . . . . . . . . . . . . . 16 ⊢ 2o = suc 1o
75	73, 74	eqeq12i 2751	. . . . . . . . . . . . . . 15 ⊢ (1o = 2o ↔ suc ∅ = suc 1o)
76		0elon 6419	. . . . . . . . . . . . . . . 16 ⊢ ∅ ∈ On
77		1on 8478	. . . . . . . . . . . . . . . 16 ⊢ 1o ∈ On
78		suc11 6472	. . . . . . . . . . . . . . . 16 ⊢ ((∅ ∈ On ∧ 1o ∈ On) → (suc ∅ = suc 1o ↔ ∅ = 1o))
79	76, 77, 78	mp2an 691	. . . . . . . . . . . . . . 15 ⊢ (suc ∅ = suc 1o ↔ ∅ = 1o)
80	75, 79	bitri 275	. . . . . . . . . . . . . 14 ⊢ (1o = 2o ↔ ∅ = 1o)
81	43, 80	nemtbir 3039	. . . . . . . . . . . . 13 ⊢ ¬ 1o = 2o
82		eqtr2 2757	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ↾ 𝑋)‘𝑥) = 1o ∧ ((𝐴 ↾ 𝑋)‘𝑥) = 2o) → 1o = 2o)
83	81, 82	mto 196	. . . . . . . . . . . 12 ⊢ ¬ (((𝐴 ↾ 𝑋)‘𝑥) = 1o ∧ ((𝐴 ↾ 𝑋)‘𝑥) = 2o)
84		2on 8480	. . . . . . . . . . . . . . . . 17 ⊢ 2o ∈ On
85	84	elexi 3494	. . . . . . . . . . . . . . . 16 ⊢ 2o ∈ V
86	85	prid2 4768	. . . . . . . . . . . . . . 15 ⊢ 2o ∈ {1o, 2o}
87	86	nosgnn0i 27162	. . . . . . . . . . . . . 14 ⊢ ∅ ≠ 2o
88	87	neii 2943	. . . . . . . . . . . . 13 ⊢ ¬ ∅ = 2o
89		eqtr2 2757	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ↾ 𝑋)‘𝑥) = ∅ ∧ ((𝐴 ↾ 𝑋)‘𝑥) = 2o) → ∅ = 2o)
90	88, 89	mto 196	. . . . . . . . . . . 12 ⊢ ¬ (((𝐴 ↾ 𝑋)‘𝑥) = ∅ ∧ ((𝐴 ↾ 𝑋)‘𝑥) = 2o)
91	72, 83, 90	3pm3.2i 1340	. . . . . . . . . . 11 ⊢ (¬ (((𝐴 ↾ 𝑋)‘𝑥) = 1o ∧ ((𝐴 ↾ 𝑋)‘𝑥) = ∅) ∧ ¬ (((𝐴 ↾ 𝑋)‘𝑥) = 1o ∧ ((𝐴 ↾ 𝑋)‘𝑥) = 2o) ∧ ¬ (((𝐴 ↾ 𝑋)‘𝑥) = ∅ ∧ ((𝐴 ↾ 𝑋)‘𝑥) = 2o))
92		fvex 6905	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ↾ 𝑋)‘𝑥) ∈ V
93	92, 92	brtp 5524	. . . . . . . . . . . . 13 ⊢ (((𝐴 ↾ 𝑋)‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} ((𝐴 ↾ 𝑋)‘𝑥) ↔ ((((𝐴 ↾ 𝑋)‘𝑥) = 1o ∧ ((𝐴 ↾ 𝑋)‘𝑥) = ∅) ∨ (((𝐴 ↾ 𝑋)‘𝑥) = 1o ∧ ((𝐴 ↾ 𝑋)‘𝑥) = 2o) ∨ (((𝐴 ↾ 𝑋)‘𝑥) = ∅ ∧ ((𝐴 ↾ 𝑋)‘𝑥) = 2o)))
94		3oran 1110	. . . . . . . . . . . . 13 ⊢ (((((𝐴 ↾ 𝑋)‘𝑥) = 1o ∧ ((𝐴 ↾ 𝑋)‘𝑥) = ∅) ∨ (((𝐴 ↾ 𝑋)‘𝑥) = 1o ∧ ((𝐴 ↾ 𝑋)‘𝑥) = 2o) ∨ (((𝐴 ↾ 𝑋)‘𝑥) = ∅ ∧ ((𝐴 ↾ 𝑋)‘𝑥) = 2o)) ↔ ¬ (¬ (((𝐴 ↾ 𝑋)‘𝑥) = 1o ∧ ((𝐴 ↾ 𝑋)‘𝑥) = ∅) ∧ ¬ (((𝐴 ↾ 𝑋)‘𝑥) = 1o ∧ ((𝐴 ↾ 𝑋)‘𝑥) = 2o) ∧ ¬ (((𝐴 ↾ 𝑋)‘𝑥) = ∅ ∧ ((𝐴 ↾ 𝑋)‘𝑥) = 2o)))
95	93, 94	bitri 275	. . . . . . . . . . . 12 ⊢ (((𝐴 ↾ 𝑋)‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} ((𝐴 ↾ 𝑋)‘𝑥) ↔ ¬ (¬ (((𝐴 ↾ 𝑋)‘𝑥) = 1o ∧ ((𝐴 ↾ 𝑋)‘𝑥) = ∅) ∧ ¬ (((𝐴 ↾ 𝑋)‘𝑥) = 1o ∧ ((𝐴 ↾ 𝑋)‘𝑥) = 2o) ∧ ¬ (((𝐴 ↾ 𝑋)‘𝑥) = ∅ ∧ ((𝐴 ↾ 𝑋)‘𝑥) = 2o)))
96	95	con2bii 358	. . . . . . . . . . 11 ⊢ ((¬ (((𝐴 ↾ 𝑋)‘𝑥) = 1o ∧ ((𝐴 ↾ 𝑋)‘𝑥) = ∅) ∧ ¬ (((𝐴 ↾ 𝑋)‘𝑥) = 1o ∧ ((𝐴 ↾ 𝑋)‘𝑥) = 2o) ∧ ¬ (((𝐴 ↾ 𝑋)‘𝑥) = ∅ ∧ ((𝐴 ↾ 𝑋)‘𝑥) = 2o)) ↔ ¬ ((𝐴 ↾ 𝑋)‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} ((𝐴 ↾ 𝑋)‘𝑥))
97	91, 96	mpbi 229	. . . . . . . . . 10 ⊢ ¬ ((𝐴 ↾ 𝑋)‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} ((𝐴 ↾ 𝑋)‘𝑥)
98		simpl2l 1227	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) ∧ (𝐵‘𝑋) = 1o) → (𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋))
99	98	adantr 482	. . . . . . . . . . . 12 ⊢ (((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) ∧ (𝐵‘𝑋) = 1o) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦))) → (𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋))
100	99	fveq1d 6894	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) ∧ (𝐵‘𝑋) = 1o) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦))) → ((𝐴 ↾ 𝑋)‘𝑥) = ((𝐵 ↾ 𝑋)‘𝑥))
101	100	breq2d 5161	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) ∧ (𝐵‘𝑋) = 1o) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦))) → (((𝐴 ↾ 𝑋)‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} ((𝐴 ↾ 𝑋)‘𝑥) ↔ ((𝐴 ↾ 𝑋)‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} ((𝐵 ↾ 𝑋)‘𝑥)))
102	97, 101	mtbii 326	. . . . . . . . 9 ⊢ (((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) ∧ (𝐵‘𝑋) = 1o) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦))) → ¬ ((𝐴 ↾ 𝑋)‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} ((𝐵 ↾ 𝑋)‘𝑥))
103		fvres 6911	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑋 → ((𝐴 ↾ 𝑋)‘𝑥) = (𝐴‘𝑥))
104		fvres 6911	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑋 → ((𝐵 ↾ 𝑋)‘𝑥) = (𝐵‘𝑥))
105	103, 104	breq12d 5162	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑋 → (((𝐴 ↾ 𝑋)‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} ((𝐵 ↾ 𝑋)‘𝑥) ↔ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑥)))
106	105	notbid 318	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑋 → (¬ ((𝐴 ↾ 𝑋)‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} ((𝐵 ↾ 𝑋)‘𝑥) ↔ ¬ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑥)))
107	102, 106	syl5ibcom 244	. . . . . . . 8 ⊢ (((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) ∧ (𝐵‘𝑋) = 1o) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦))) → (𝑥 ∈ 𝑋 → ¬ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑥)))
108	44	intnanr 489	. . . . . . . . . . . 12 ⊢ ¬ (∅ = 1o ∧ 1o = ∅)
109	44	intnanr 489	. . . . . . . . . . . 12 ⊢ ¬ (∅ = 1o ∧ 1o = 2o)
110	81	intnan 488	. . . . . . . . . . . 12 ⊢ ¬ (∅ = ∅ ∧ 1o = 2o)
111	108, 109, 110	3pm3.2i 1340	. . . . . . . . . . 11 ⊢ (¬ (∅ = 1o ∧ 1o = ∅) ∧ ¬ (∅ = 1o ∧ 1o = 2o) ∧ ¬ (∅ = ∅ ∧ 1o = 2o))
112		0ex 5308	. . . . . . . . . . . . . 14 ⊢ ∅ ∈ V
113	112, 41	brtp 5524	. . . . . . . . . . . . 13 ⊢ (∅{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}1o ↔ ((∅ = 1o ∧ 1o = ∅) ∨ (∅ = 1o ∧ 1o = 2o) ∨ (∅ = ∅ ∧ 1o = 2o)))
114		3oran 1110	. . . . . . . . . . . . 13 ⊢ (((∅ = 1o ∧ 1o = ∅) ∨ (∅ = 1o ∧ 1o = 2o) ∨ (∅ = ∅ ∧ 1o = 2o)) ↔ ¬ (¬ (∅ = 1o ∧ 1o = ∅) ∧ ¬ (∅ = 1o ∧ 1o = 2o) ∧ ¬ (∅ = ∅ ∧ 1o = 2o)))
115	113, 114	bitri 275	. . . . . . . . . . . 12 ⊢ (∅{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}1o ↔ ¬ (¬ (∅ = 1o ∧ 1o = ∅) ∧ ¬ (∅ = 1o ∧ 1o = 2o) ∧ ¬ (∅ = ∅ ∧ 1o = 2o)))
116	115	con2bii 358	. . . . . . . . . . 11 ⊢ ((¬ (∅ = 1o ∧ 1o = ∅) ∧ ¬ (∅ = 1o ∧ 1o = 2o) ∧ ¬ (∅ = ∅ ∧ 1o = 2o)) ↔ ¬ ∅{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}1o)
117	111, 116	mpbi 229	. . . . . . . . . 10 ⊢ ¬ ∅{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}1o
118	45, 46	breq12d 5162	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) ∧ (𝐵‘𝑋) = 1o) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦))) → ((𝐴‘𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑋) ↔ ∅{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}1o))
119	117, 118	mtbiri 327	. . . . . . . . 9 ⊢ (((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) ∧ (𝐵‘𝑋) = 1o) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦))) → ¬ (𝐴‘𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑋))
120		fveq2 6892	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (𝐴‘𝑥) = (𝐴‘𝑋))
121		fveq2 6892	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (𝐵‘𝑥) = (𝐵‘𝑋))
122	120, 121	breq12d 5162	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → ((𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑥) ↔ (𝐴‘𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑋)))
123	122	notbid 318	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (¬ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑥) ↔ ¬ (𝐴‘𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑋)))
124	119, 123	syl5ibrcom 246	. . . . . . . 8 ⊢ (((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) ∧ (𝐵‘𝑋) = 1o) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦))) → (𝑥 = 𝑋 → ¬ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑥)))
125	107, 124	jaod 858	. . . . . . 7 ⊢ (((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) ∧ (𝐵‘𝑋) = 1o) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦))) → ((𝑥 ∈ 𝑋 ∨ 𝑥 = 𝑋) → ¬ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑥)))
126	69, 125	sylbid 239	. . . . . 6 ⊢ (((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) ∧ (𝐵‘𝑋) = 1o) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦))) → (𝑥 ⊆ 𝑋 → ¬ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑥)))
127	67, 126	mpd 15	. . . . 5 ⊢ (((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) ∧ (𝐵‘𝑋) = 1o) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦))) → ¬ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑥))
128	127	expr 458	. . . 4 ⊢ (((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) ∧ (𝐵‘𝑋) = 1o) ∧ 𝑥 ∈ On) → (∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦) → ¬ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑥)))
129	128	ralrimiva 3147	. . 3 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) ∧ (𝐵‘𝑋) = 1o) → ∀𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦) → ¬ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑥)))
130	40, 129	mtand 815	. 2 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) → ¬ (𝐵‘𝑋) = 1o)
131		nofv 27160	. . . 4 ⊢ (𝐵 ∈ No → ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 1o ∨ (𝐵‘𝑋) = 2o))
132	32, 131	syl 17	. . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) → ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 1o ∨ (𝐵‘𝑋) = 2o))
133		3orrot 1093	. . . 4 ⊢ (((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 1o ∨ (𝐵‘𝑋) = 2o) ↔ ((𝐵‘𝑋) = 1o ∨ (𝐵‘𝑋) = 2o ∨ (𝐵‘𝑋) = ∅))
134		3orrot 1093	. . . 4 ⊢ (((𝐵‘𝑋) = 1o ∨ (𝐵‘𝑋) = 2o ∨ (𝐵‘𝑋) = ∅) ↔ ((𝐵‘𝑋) = 2o ∨ (𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 1o))
135	133, 134	bitri 275	. . 3 ⊢ (((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 1o ∨ (𝐵‘𝑋) = 2o) ↔ ((𝐵‘𝑋) = 2o ∨ (𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 1o))
136	132, 135	sylib 217	. 2 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) → ((𝐵‘𝑋) = 2o ∨ (𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 1o))
137	31, 130, 136	ecase23d 1474	1 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) → (𝐵‘𝑋) = 2o)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   ∨ w3o 1087   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  ∀wral 3062  ∃wrex 3071   ⊆ wss 3949  ∅c0 4323  {ctp 4633  ⟨cop 4635   class class class wbr 5149   Or wor 5588  dom cdm 5677   ↾ cres 5679  Rel wrel 5682  Oncon0 6365  suc csuc 6367  Fun wfun 6538  ‘cfv 6544  1_oc1o 8459  2_oc2o 8460   No csur 27143   <s cslt 27144

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-1o 8466  df-2o 8467  df-no 27146  df-slt 27147

This theorem is referenced by:  nosupbnd1lem4  27214