Theorem nogt01o 27741

Description: Given 𝐴 greater than 𝐵, equal to 𝐵 up to 𝑋, and 𝐵(𝑋) undefined, then 𝐴(𝑋) = 1_o. (Contributed by Scott Fenton, 9-Aug-2024.)

Assertion

Ref	Expression
nogt01o	⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵‘𝑋) = ∅) → (𝐴‘𝑋) = 1o)

Proof of Theorem nogt01o

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

Step	Hyp	Ref	Expression
1		sltso 27721	. . . 4 ⊢ <s Or No
2		simp11 1204	. . . 4 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵‘𝑋) = ∅) → 𝐴 ∈ No )
3		sonr 5616	. . . 4 ⊢ (( <s Or No ∧ 𝐴 ∈ No ) → ¬ 𝐴 <s 𝐴)
4	1, 2, 3	sylancr 587	. . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵‘𝑋) = ∅) → ¬ 𝐴 <s 𝐴)
5		simpl2r 1228	. . . 4 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵‘𝑋) = ∅) ∧ (𝐴‘𝑋) = ∅) → 𝐴 <s 𝐵)
6		simpl2l 1227	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵‘𝑋) = ∅) ∧ (𝐴‘𝑋) = ∅) → (𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋))
7		simpl11 1249	. . . . . . 7 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵‘𝑋) = ∅) ∧ (𝐴‘𝑋) = ∅) → 𝐴 ∈ No )
8		nofun 27694	. . . . . . . 8 ⊢ (𝐴 ∈ No → Fun 𝐴)
9		funrel 6583	. . . . . . . 8 ⊢ (Fun 𝐴 → Rel 𝐴)
10	8, 9	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ No → Rel 𝐴)
11	7, 10	syl 17	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵‘𝑋) = ∅) ∧ (𝐴‘𝑋) = ∅) → Rel 𝐴)
12		simpl13 1251	. . . . . . 7 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵‘𝑋) = ∅) ∧ (𝐴‘𝑋) = ∅) → 𝑋 ∈ On)
13		simpr 484	. . . . . . 7 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵‘𝑋) = ∅) ∧ (𝐴‘𝑋) = ∅) → (𝐴‘𝑋) = ∅)
14		nolt02olem 27739	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑋 ∈ On ∧ (𝐴‘𝑋) = ∅) → dom 𝐴 ⊆ 𝑋)
15	7, 12, 13, 14	syl3anc 1373	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵‘𝑋) = ∅) ∧ (𝐴‘𝑋) = ∅) → dom 𝐴 ⊆ 𝑋)
16		relssres 6040	. . . . . 6 ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ 𝑋) → (𝐴 ↾ 𝑋) = 𝐴)
17	11, 15, 16	syl2anc 584	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵‘𝑋) = ∅) ∧ (𝐴‘𝑋) = ∅) → (𝐴 ↾ 𝑋) = 𝐴)
18		simpl12 1250	. . . . . . 7 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵‘𝑋) = ∅) ∧ (𝐴‘𝑋) = ∅) → 𝐵 ∈ No )
19		nofun 27694	. . . . . . . 8 ⊢ (𝐵 ∈ No → Fun 𝐵)
20		funrel 6583	. . . . . . . 8 ⊢ (Fun 𝐵 → Rel 𝐵)
21	19, 20	syl 17	. . . . . . 7 ⊢ (𝐵 ∈ No → Rel 𝐵)
22	18, 21	syl 17	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵‘𝑋) = ∅) ∧ (𝐴‘𝑋) = ∅) → Rel 𝐵)
23		simpl3 1194	. . . . . . 7 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵‘𝑋) = ∅) ∧ (𝐴‘𝑋) = ∅) → (𝐵‘𝑋) = ∅)
24		nolt02olem 27739	. . . . . . 7 ⊢ ((𝐵 ∈ No ∧ 𝑋 ∈ On ∧ (𝐵‘𝑋) = ∅) → dom 𝐵 ⊆ 𝑋)
25	18, 12, 23, 24	syl3anc 1373	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵‘𝑋) = ∅) ∧ (𝐴‘𝑋) = ∅) → dom 𝐵 ⊆ 𝑋)
26		relssres 6040	. . . . . 6 ⊢ ((Rel 𝐵 ∧ dom 𝐵 ⊆ 𝑋) → (𝐵 ↾ 𝑋) = 𝐵)
27	22, 25, 26	syl2anc 584	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵‘𝑋) = ∅) ∧ (𝐴‘𝑋) = ∅) → (𝐵 ↾ 𝑋) = 𝐵)
28	6, 17, 27	3eqtr3d 2785	. . . 4 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵‘𝑋) = ∅) ∧ (𝐴‘𝑋) = ∅) → 𝐴 = 𝐵)
29	5, 28	breqtrrd 5171	. . 3 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵‘𝑋) = ∅) ∧ (𝐴‘𝑋) = ∅) → 𝐴 <s 𝐴)
30	4, 29	mtand 816	. 2 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵‘𝑋) = ∅) → ¬ (𝐴‘𝑋) = ∅)
31		simp2r 1201	. . . . 5 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵‘𝑋) = ∅) → 𝐴 <s 𝐵)
32		simp12 1205	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵‘𝑋) = ∅) → 𝐵 ∈ No )
33		sltval 27692	. . . . . 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	2, 32, 33	syl2anc 584	. . . . 5 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵‘𝑋) = ∅) → (𝐴 <s 𝐵 ↔ ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦) ∧ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑥))))
35	31, 34	mpbid 232	. . . 4 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵‘𝑋) = ∅) → ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦) ∧ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑥)))
36		ralinexa 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)))
37	36	con2bii 357	. . . 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)))
38	35, 37	sylib 218	. . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵‘𝑋) = ∅) → ¬ ∀𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦) → ¬ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑥)))
39		1n0 8526	. . . . . . . . . . . 12 ⊢ 1o ≠ ∅
40	39	neii 2942	. . . . . . . . . . 11 ⊢ ¬ 1o = ∅
41		eqtr2 2761	. . . . . . . . . . 11 ⊢ ((((𝐴 ↾ 𝑋)‘𝑥) = 1o ∧ ((𝐴 ↾ 𝑋)‘𝑥) = ∅) → 1o = ∅)
42	40, 41	mto 197	. . . . . . . . . 10 ⊢ ¬ (((𝐴 ↾ 𝑋)‘𝑥) = 1o ∧ ((𝐴 ↾ 𝑋)‘𝑥) = ∅)
43		df-2o 8507	. . . . . . . . . . . . 13 ⊢ 2o = suc 1o
44		2on 8520	. . . . . . . . . . . . . . . 16 ⊢ 2o ∈ On
45	43, 44	eqeltrri 2838	. . . . . . . . . . . . . . 15 ⊢ suc 1o ∈ On
46	45	onordi 6495	. . . . . . . . . . . . . 14 ⊢ Ord suc 1o
47		1oex 8516	. . . . . . . . . . . . . . 15 ⊢ 1o ∈ V
48	47	sucid 6466	. . . . . . . . . . . . . 14 ⊢ 1o ∈ suc 1o
49		nordeq 6403	. . . . . . . . . . . . . 14 ⊢ ((Ord suc 1o ∧ 1o ∈ suc 1o) → suc 1o ≠ 1o)
50	46, 48, 49	mp2an 692	. . . . . . . . . . . . 13 ⊢ suc 1o ≠ 1o
51	43, 50	eqnetri 3011	. . . . . . . . . . . 12 ⊢ 2o ≠ 1o
52	51	nesymi 2998	. . . . . . . . . . 11 ⊢ ¬ 1o = 2o
53		eqtr2 2761	. . . . . . . . . . 11 ⊢ ((((𝐴 ↾ 𝑋)‘𝑥) = 1o ∧ ((𝐴 ↾ 𝑋)‘𝑥) = 2o) → 1o = 2o)
54	52, 53	mto 197	. . . . . . . . . 10 ⊢ ¬ (((𝐴 ↾ 𝑋)‘𝑥) = 1o ∧ ((𝐴 ↾ 𝑋)‘𝑥) = 2o)
55		2on0 8522	. . . . . . . . . . . 12 ⊢ 2o ≠ ∅
56	55	nesymi 2998	. . . . . . . . . . 11 ⊢ ¬ ∅ = 2o
57		eqtr2 2761	. . . . . . . . . . 11 ⊢ ((((𝐴 ↾ 𝑋)‘𝑥) = ∅ ∧ ((𝐴 ↾ 𝑋)‘𝑥) = 2o) → ∅ = 2o)
58	56, 57	mto 197	. . . . . . . . . 10 ⊢ ¬ (((𝐴 ↾ 𝑋)‘𝑥) = ∅ ∧ ((𝐴 ↾ 𝑋)‘𝑥) = 2o)
59	42, 54, 58	3pm3.2i 1340	. . . . . . . . 9 ⊢ (¬ (((𝐴 ↾ 𝑋)‘𝑥) = 1o ∧ ((𝐴 ↾ 𝑋)‘𝑥) = ∅) ∧ ¬ (((𝐴 ↾ 𝑋)‘𝑥) = 1o ∧ ((𝐴 ↾ 𝑋)‘𝑥) = 2o) ∧ ¬ (((𝐴 ↾ 𝑋)‘𝑥) = ∅ ∧ ((𝐴 ↾ 𝑋)‘𝑥) = 2o))
60		fvex 6919	. . . . . . . . . . . 12 ⊢ ((𝐴 ↾ 𝑋)‘𝑥) ∈ V
61	60, 60	brtp 5528	. . . . . . . . . . 11 ⊢ (((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)))
62		3oran 1109	. . . . . . . . . . 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 ∧ ((𝐴 ↾ 𝑋)‘𝑥) = ∅) ∧ ¬ (((𝐴 ↾ 𝑋)‘𝑥) = 1o ∧ ((𝐴 ↾ 𝑋)‘𝑥) = 2o) ∧ ¬ (((𝐴 ↾ 𝑋)‘𝑥) = ∅ ∧ ((𝐴 ↾ 𝑋)‘𝑥) = 2o)))
63	61, 62	bitri 275	. . . . . . . . . 10 ⊢ (((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)))
64	63	con2bii 357	. . . . . . . . 9 ⊢ ((¬ (((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))
65	59, 64	mpbi 230	. . . . . . . 8 ⊢ ¬ ((𝐴 ↾ 𝑋)‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} ((𝐴 ↾ 𝑋)‘𝑥)
66		simpl2l 1227	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵‘𝑋) = ∅) ∧ (𝐴‘𝑋) = 2o) → (𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋))
67	66	adantr 480	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵‘𝑋) = ∅) ∧ (𝐴‘𝑋) = 2o) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦))) → (𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋))
68	67	fveq1d 6908	. . . . . . . . 9 ⊢ (((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵‘𝑋) = ∅) ∧ (𝐴‘𝑋) = 2o) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦))) → ((𝐴 ↾ 𝑋)‘𝑥) = ((𝐵 ↾ 𝑋)‘𝑥))
69	68	breq2d 5155	. . . . . . . 8 ⊢ (((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵‘𝑋) = ∅) ∧ (𝐴‘𝑋) = 2o) ∧ (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)))
70	65, 69	mtbii 326	. . . . . . 7 ⊢ (((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵‘𝑋) = ∅) ∧ (𝐴‘𝑋) = 2o) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦))) → ¬ ((𝐴 ↾ 𝑋)‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} ((𝐵 ↾ 𝑋)‘𝑥))
71		fvres 6925	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑋 → ((𝐴 ↾ 𝑋)‘𝑥) = (𝐴‘𝑥))
72		fvres 6925	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑋 → ((𝐵 ↾ 𝑋)‘𝑥) = (𝐵‘𝑥))
73	71, 72	breq12d 5156	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑋 → (((𝐴 ↾ 𝑋)‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} ((𝐵 ↾ 𝑋)‘𝑥) ↔ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑥)))
74	73	notbid 318	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋 → (¬ ((𝐴 ↾ 𝑋)‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} ((𝐵 ↾ 𝑋)‘𝑥) ↔ ¬ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑥)))
75	70, 74	syl5ibcom 245	. . . . . 6 ⊢ (((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵‘𝑋) = ∅) ∧ (𝐴‘𝑋) = 2o) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦))) → (𝑥 ∈ 𝑋 → ¬ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑥)))
76	51	neii 2942	. . . . . . . . . . 11 ⊢ ¬ 2o = 1o
77	76	intnanr 487	. . . . . . . . . 10 ⊢ ¬ (2o = 1o ∧ ∅ = ∅)
78	56	intnan 486	. . . . . . . . . 10 ⊢ ¬ (2o = 1o ∧ ∅ = 2o)
79	56	intnan 486	. . . . . . . . . 10 ⊢ ¬ (2o = ∅ ∧ ∅ = 2o)
80	77, 78, 79	3pm3.2i 1340	. . . . . . . . 9 ⊢ (¬ (2o = 1o ∧ ∅ = ∅) ∧ ¬ (2o = 1o ∧ ∅ = 2o) ∧ ¬ (2o = ∅ ∧ ∅ = 2o))
81		2oex 8517	. . . . . . . . . . . 12 ⊢ 2o ∈ V
82		0ex 5307	. . . . . . . . . . . 12 ⊢ ∅ ∈ V
83	81, 82	brtp 5528	. . . . . . . . . . 11 ⊢ (2o{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}∅ ↔ ((2o = 1o ∧ ∅ = ∅) ∨ (2o = 1o ∧ ∅ = 2o) ∨ (2o = ∅ ∧ ∅ = 2o)))
84		3oran 1109	. . . . . . . . . . 11 ⊢ (((2o = 1o ∧ ∅ = ∅) ∨ (2o = 1o ∧ ∅ = 2o) ∨ (2o = ∅ ∧ ∅ = 2o)) ↔ ¬ (¬ (2o = 1o ∧ ∅ = ∅) ∧ ¬ (2o = 1o ∧ ∅ = 2o) ∧ ¬ (2o = ∅ ∧ ∅ = 2o)))
85	83, 84	bitri 275	. . . . . . . . . 10 ⊢ (2o{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}∅ ↔ ¬ (¬ (2o = 1o ∧ ∅ = ∅) ∧ ¬ (2o = 1o ∧ ∅ = 2o) ∧ ¬ (2o = ∅ ∧ ∅ = 2o)))
86	85	con2bii 357	. . . . . . . . 9 ⊢ ((¬ (2o = 1o ∧ ∅ = ∅) ∧ ¬ (2o = 1o ∧ ∅ = 2o) ∧ ¬ (2o = ∅ ∧ ∅ = 2o)) ↔ ¬ 2o{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}∅)
87	80, 86	mpbi 230	. . . . . . . 8 ⊢ ¬ 2o{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}∅
88		simplr 769	. . . . . . . . 9 ⊢ (((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵‘𝑋) = ∅) ∧ (𝐴‘𝑋) = 2o) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦))) → (𝐴‘𝑋) = 2o)
89		simpll3 1215	. . . . . . . . 9 ⊢ (((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵‘𝑋) = ∅) ∧ (𝐴‘𝑋) = 2o) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦))) → (𝐵‘𝑋) = ∅)
90	88, 89	breq12d 5156	. . . . . . . 8 ⊢ (((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵‘𝑋) = ∅) ∧ (𝐴‘𝑋) = 2o) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦))) → ((𝐴‘𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑋) ↔ 2o{⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩}∅))
91	87, 90	mtbiri 327	. . . . . . 7 ⊢ (((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵‘𝑋) = ∅) ∧ (𝐴‘𝑋) = 2o) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦))) → ¬ (𝐴‘𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑋))
92		fveq2 6906	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝐴‘𝑥) = (𝐴‘𝑋))
93		fveq2 6906	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝐵‘𝑥) = (𝐵‘𝑋))
94	92, 93	breq12d 5156	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ((𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑥) ↔ (𝐴‘𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑋)))
95	94	notbid 318	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (¬ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑥) ↔ ¬ (𝐴‘𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑋)))
96	91, 95	syl5ibrcom 247	. . . . . 6 ⊢ (((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵‘𝑋) = ∅) ∧ (𝐴‘𝑋) = 2o) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦))) → (𝑥 = 𝑋 → ¬ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑥)))
97		fveq2 6906	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑋 → (𝐴‘𝑦) = (𝐴‘𝑋))
98		fveq2 6906	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑋 → (𝐵‘𝑦) = (𝐵‘𝑋))
99	97, 98	eqeq12d 2753	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑋 → ((𝐴‘𝑦) = (𝐵‘𝑦) ↔ (𝐴‘𝑋) = (𝐵‘𝑋)))
100	99	rspccv 3619	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦) → (𝑋 ∈ 𝑥 → (𝐴‘𝑋) = (𝐵‘𝑋)))
101	100	ad2antll 729	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵‘𝑋) = ∅) ∧ (𝐴‘𝑋) = 2o) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦))) → (𝑋 ∈ 𝑥 → (𝐴‘𝑋) = (𝐵‘𝑋)))
102		eqcom 2744	. . . . . . . . . 10 ⊢ ((𝐴‘𝑋) = (𝐵‘𝑋) ↔ (𝐵‘𝑋) = (𝐴‘𝑋))
103	101, 102	imbitrdi 251	. . . . . . . . 9 ⊢ (((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵‘𝑋) = ∅) ∧ (𝐴‘𝑋) = 2o) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦))) → (𝑋 ∈ 𝑥 → (𝐵‘𝑋) = (𝐴‘𝑋)))
104	89, 88	eqeq12d 2753	. . . . . . . . 9 ⊢ (((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵‘𝑋) = ∅) ∧ (𝐴‘𝑋) = 2o) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦))) → ((𝐵‘𝑋) = (𝐴‘𝑋) ↔ ∅ = 2o))
105	103, 104	sylibd 239	. . . . . . . 8 ⊢ (((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵‘𝑋) = ∅) ∧ (𝐴‘𝑋) = 2o) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦))) → (𝑋 ∈ 𝑥 → ∅ = 2o))
106	56, 105	mtoi 199	. . . . . . 7 ⊢ (((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵‘𝑋) = ∅) ∧ (𝐴‘𝑋) = 2o) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦))) → ¬ 𝑋 ∈ 𝑥)
107		simprl 771	. . . . . . . 8 ⊢ (((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵‘𝑋) = ∅) ∧ (𝐴‘𝑋) = 2o) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦))) → 𝑥 ∈ On)
108		simpl13 1251	. . . . . . . . 9 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵‘𝑋) = ∅) ∧ (𝐴‘𝑋) = 2o) → 𝑋 ∈ On)
109	108	adantr 480	. . . . . . . 8 ⊢ (((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵‘𝑋) = ∅) ∧ (𝐴‘𝑋) = 2o) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦))) → 𝑋 ∈ On)
110		ontri1 6418	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ 𝑋 ∈ On) → (𝑥 ⊆ 𝑋 ↔ ¬ 𝑋 ∈ 𝑥))
111		onsseleq 6425	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ 𝑋 ∈ On) → (𝑥 ⊆ 𝑋 ↔ (𝑥 ∈ 𝑋 ∨ 𝑥 = 𝑋)))
112	110, 111	bitr3d 281	. . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ 𝑋 ∈ On) → (¬ 𝑋 ∈ 𝑥 ↔ (𝑥 ∈ 𝑋 ∨ 𝑥 = 𝑋)))
113	107, 109, 112	syl2anc 584	. . . . . . 7 ⊢ (((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵‘𝑋) = ∅) ∧ (𝐴‘𝑋) = 2o) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦))) → (¬ 𝑋 ∈ 𝑥 ↔ (𝑥 ∈ 𝑋 ∨ 𝑥 = 𝑋)))
114	106, 113	mpbid 232	. . . . . 6 ⊢ (((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵‘𝑋) = ∅) ∧ (𝐴‘𝑋) = 2o) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦))) → (𝑥 ∈ 𝑋 ∨ 𝑥 = 𝑋))
115	75, 96, 114	mpjaod 861	. . . . 5 ⊢ (((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵‘𝑋) = ∅) ∧ (𝐴‘𝑋) = 2o) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦))) → ¬ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑥))
116	115	expr 456	. . . 4 ⊢ (((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵‘𝑋) = ∅) ∧ (𝐴‘𝑋) = 2o) ∧ 𝑥 ∈ On) → (∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦) → ¬ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑥)))
117	116	ralrimiva 3146	. . 3 ⊢ ((((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵‘𝑋) = ∅) ∧ (𝐴‘𝑋) = 2o) → ∀𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦) → ¬ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑥)))
118	38, 117	mtand 816	. 2 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵‘𝑋) = ∅) → ¬ (𝐴‘𝑋) = 2o)
119		nofv 27702	. . . 4 ⊢ (𝐴 ∈ No → ((𝐴‘𝑋) = ∅ ∨ (𝐴‘𝑋) = 1o ∨ (𝐴‘𝑋) = 2o))
120	2, 119	syl 17	. . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵‘𝑋) = ∅) → ((𝐴‘𝑋) = ∅ ∨ (𝐴‘𝑋) = 1o ∨ (𝐴‘𝑋) = 2o))
121		3orcoma 1093	. . 3 ⊢ (((𝐴‘𝑋) = ∅ ∨ (𝐴‘𝑋) = 1o ∨ (𝐴‘𝑋) = 2o) ↔ ((𝐴‘𝑋) = 1o ∨ (𝐴‘𝑋) = ∅ ∨ (𝐴‘𝑋) = 2o))
122	120, 121	sylib 218	. 2 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵‘𝑋) = ∅) → ((𝐴‘𝑋) = 1o ∨ (𝐴‘𝑋) = ∅ ∨ (𝐴‘𝑋) = 2o))
123	30, 118, 122	ecase23d 1475	1 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵‘𝑋) = ∅) → (𝐴‘𝑋) = 1o)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∨ w3o 1086   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108   ≠ wne 2940  ∀wral 3061  ∃wrex 3070   ⊆ wss 3951  ∅c0 4333  {ctp 4630  ⟨cop 4632   class class class wbr 5143   Or wor 5591  dom cdm 5685   ↾ cres 5687  Rel wrel 5690  Ord word 6383  Oncon0 6384  suc csuc 6386  Fun wfun 6555  ‘cfv 6561  1_oc1o 8499  2_oc2o 8500   No csur 27684   <s cslt 27685

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-1o 8506  df-2o 8507  df-no 27687  df-slt 27688

This theorem is referenced by:  noinfbnd1lem4  27771