Theorem nolesgn2o 32161

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

Assertion

Ref	Expression
nolesgn2o	⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2𝑜) ∧ ¬ 𝐵 <s 𝐴) → (𝐵‘𝑋) = 2𝑜)

Proof of Theorem nolesgn2o

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

Step	Hyp	Ref	Expression
1		simpl2 1229	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2𝑜)) → 𝐵 ∈ No )
2		nofv 32147	. . . . . 6 ⊢ (𝐵 ∈ No → ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 1𝑜 ∨ (𝐵‘𝑋) = 2𝑜))
3	1, 2	syl 17	. . . . 5 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2𝑜)) → ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 1𝑜 ∨ (𝐵‘𝑋) = 2𝑜))
4		3orel3 31931	. . . . 5 ⊢ (¬ (𝐵‘𝑋) = 2𝑜 → (((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 1𝑜 ∨ (𝐵‘𝑋) = 2𝑜) → ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 1𝑜)))
5	3, 4	syl5com 31	. . . 4 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2𝑜)) → (¬ (𝐵‘𝑋) = 2𝑜 → ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 1𝑜)))
6		simp13 1247	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2𝑜) ∧ ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 1𝑜)) → 𝑋 ∈ On)
7		fveq1 6331	. . . . . . . . . . . . 13 ⊢ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) → ((𝐴 ↾ 𝑋)‘𝑦) = ((𝐵 ↾ 𝑋)‘𝑦))
8	7	eqcomd 2777	. . . . . . . . . . . 12 ⊢ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) → ((𝐵 ↾ 𝑋)‘𝑦) = ((𝐴 ↾ 𝑋)‘𝑦))
9	8	adantr 466	. . . . . . . . . . 11 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝐵 ↾ 𝑋)‘𝑦) = ((𝐴 ↾ 𝑋)‘𝑦))
10		simpr 471	. . . . . . . . . . . 12 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ 𝑋)
11	10	fvresd 6349	. . . . . . . . . . 11 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝐵 ↾ 𝑋)‘𝑦) = (𝐵‘𝑦))
12	10	fvresd 6349	. . . . . . . . . . 11 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝐴 ↾ 𝑋)‘𝑦) = (𝐴‘𝑦))
13	9, 11, 12	3eqtr3d 2813	. . . . . . . . . 10 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝑦 ∈ 𝑋) → (𝐵‘𝑦) = (𝐴‘𝑦))
14	13	ralrimiva 3115	. . . . . . . . 9 ⊢ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) → ∀𝑦 ∈ 𝑋 (𝐵‘𝑦) = (𝐴‘𝑦))
15	14	adantr 466	. . . . . . . 8 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2𝑜) → ∀𝑦 ∈ 𝑋 (𝐵‘𝑦) = (𝐴‘𝑦))
16	15	3ad2ant2 1128	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2𝑜) ∧ ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 1𝑜)) → ∀𝑦 ∈ 𝑋 (𝐵‘𝑦) = (𝐴‘𝑦))
17		simprr 756	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2𝑜)) → (𝐴‘𝑋) = 2𝑜)
18	17	a1d 25	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2𝑜)) → ((𝐵‘𝑋) = ∅ → (𝐴‘𝑋) = 2𝑜))
19	18	ancld 540	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2𝑜)) → ((𝐵‘𝑋) = ∅ → ((𝐵‘𝑋) = ∅ ∧ (𝐴‘𝑋) = 2𝑜)))
20	17	a1d 25	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2𝑜)) → ((𝐵‘𝑋) = 1𝑜 → (𝐴‘𝑋) = 2𝑜))
21	20	ancld 540	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2𝑜)) → ((𝐵‘𝑋) = 1𝑜 → ((𝐵‘𝑋) = 1𝑜 ∧ (𝐴‘𝑋) = 2𝑜)))
22	19, 21	orim12d 945	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2𝑜)) → (((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 1𝑜) → (((𝐵‘𝑋) = ∅ ∧ (𝐴‘𝑋) = 2𝑜) ∨ ((𝐵‘𝑋) = 1𝑜 ∧ (𝐴‘𝑋) = 2𝑜))))
23	22	3impia 1109	. . . . . . . . 9 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2𝑜) ∧ ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 1𝑜)) → (((𝐵‘𝑋) = ∅ ∧ (𝐴‘𝑋) = 2𝑜) ∨ ((𝐵‘𝑋) = 1𝑜 ∧ (𝐴‘𝑋) = 2𝑜)))
24		3mix3 1416	. . . . . . . . . 10 ⊢ (((𝐵‘𝑋) = ∅ ∧ (𝐴‘𝑋) = 2𝑜) → (((𝐵‘𝑋) = 1𝑜 ∧ (𝐴‘𝑋) = ∅) ∨ ((𝐵‘𝑋) = 1𝑜 ∧ (𝐴‘𝑋) = 2𝑜) ∨ ((𝐵‘𝑋) = ∅ ∧ (𝐴‘𝑋) = 2𝑜)))
25		3mix2 1415	. . . . . . . . . 10 ⊢ (((𝐵‘𝑋) = 1𝑜 ∧ (𝐴‘𝑋) = 2𝑜) → (((𝐵‘𝑋) = 1𝑜 ∧ (𝐴‘𝑋) = ∅) ∨ ((𝐵‘𝑋) = 1𝑜 ∧ (𝐴‘𝑋) = 2𝑜) ∨ ((𝐵‘𝑋) = ∅ ∧ (𝐴‘𝑋) = 2𝑜)))
26	24, 25	jaoi 844	. . . . . . . . 9 ⊢ ((((𝐵‘𝑋) = ∅ ∧ (𝐴‘𝑋) = 2𝑜) ∨ ((𝐵‘𝑋) = 1𝑜 ∧ (𝐴‘𝑋) = 2𝑜)) → (((𝐵‘𝑋) = 1𝑜 ∧ (𝐴‘𝑋) = ∅) ∨ ((𝐵‘𝑋) = 1𝑜 ∧ (𝐴‘𝑋) = 2𝑜) ∨ ((𝐵‘𝑋) = ∅ ∧ (𝐴‘𝑋) = 2𝑜)))
27	23, 26	syl 17	. . . . . . . 8 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2𝑜) ∧ ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 1𝑜)) → (((𝐵‘𝑋) = 1𝑜 ∧ (𝐴‘𝑋) = ∅) ∨ ((𝐵‘𝑋) = 1𝑜 ∧ (𝐴‘𝑋) = 2𝑜) ∨ ((𝐵‘𝑋) = ∅ ∧ (𝐴‘𝑋) = 2𝑜)))
28		fvex 6342	. . . . . . . . 9 ⊢ (𝐵‘𝑋) ∈ V
29		fvex 6342	. . . . . . . . 9 ⊢ (𝐴‘𝑋) ∈ V
30	28, 29	brtp 31977	. . . . . . . 8 ⊢ ((𝐵‘𝑋){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐴‘𝑋) ↔ (((𝐵‘𝑋) = 1𝑜 ∧ (𝐴‘𝑋) = ∅) ∨ ((𝐵‘𝑋) = 1𝑜 ∧ (𝐴‘𝑋) = 2𝑜) ∨ ((𝐵‘𝑋) = ∅ ∧ (𝐴‘𝑋) = 2𝑜)))
31	27, 30	sylibr 224	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2𝑜) ∧ ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 1𝑜)) → (𝐵‘𝑋){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐴‘𝑋))
32		raleq 3287	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (∀𝑦 ∈ 𝑥 (𝐵‘𝑦) = (𝐴‘𝑦) ↔ ∀𝑦 ∈ 𝑋 (𝐵‘𝑦) = (𝐴‘𝑦)))
33		fveq2 6332	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝐵‘𝑥) = (𝐵‘𝑋))
34		fveq2 6332	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝐴‘𝑥) = (𝐴‘𝑋))
35	33, 34	breq12d 4799	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → ((𝐵‘𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐴‘𝑥) ↔ (𝐵‘𝑋){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐴‘𝑋)))
36	32, 35	anbi12d 616	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ((∀𝑦 ∈ 𝑥 (𝐵‘𝑦) = (𝐴‘𝑦) ∧ (𝐵‘𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐴‘𝑥)) ↔ (∀𝑦 ∈ 𝑋 (𝐵‘𝑦) = (𝐴‘𝑦) ∧ (𝐵‘𝑋){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐴‘𝑋))))
37	36	rspcev 3460	. . . . . . 7 ⊢ ((𝑋 ∈ On ∧ (∀𝑦 ∈ 𝑋 (𝐵‘𝑦) = (𝐴‘𝑦) ∧ (𝐵‘𝑋){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐴‘𝑋))) → ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐵‘𝑦) = (𝐴‘𝑦) ∧ (𝐵‘𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐴‘𝑥)))
38	6, 16, 31, 37	syl12anc 1474	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2𝑜) ∧ ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 1𝑜)) → ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐵‘𝑦) = (𝐴‘𝑦) ∧ (𝐵‘𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐴‘𝑥)))
39		simp12 1246	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2𝑜) ∧ ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 1𝑜)) → 𝐵 ∈ No )
40		simp11 1245	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2𝑜) ∧ ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 1𝑜)) → 𝐴 ∈ No )
41		sltval 32137	. . . . . . 7 ⊢ ((𝐵 ∈ No ∧ 𝐴 ∈ No ) → (𝐵 <s 𝐴 ↔ ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐵‘𝑦) = (𝐴‘𝑦) ∧ (𝐵‘𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐴‘𝑥))))
42	39, 40, 41	syl2anc 573	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2𝑜) ∧ ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 1𝑜)) → (𝐵 <s 𝐴 ↔ ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐵‘𝑦) = (𝐴‘𝑦) ∧ (𝐵‘𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐴‘𝑥))))
43	38, 42	mpbird 247	. . . . 5 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2𝑜) ∧ ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 1𝑜)) → 𝐵 <s 𝐴)
44	43	3expia 1114	. . . 4 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2𝑜)) → (((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 1𝑜) → 𝐵 <s 𝐴))
45	5, 44	syld 47	. . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2𝑜)) → (¬ (𝐵‘𝑋) = 2𝑜 → 𝐵 <s 𝐴))
46	45	con1d 141	. 2 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2𝑜)) → (¬ 𝐵 <s 𝐴 → (𝐵‘𝑋) = 2𝑜))
47	46	3impia 1109	1 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2𝑜) ∧ ¬ 𝐵 <s 𝐴) → (𝐵‘𝑋) = 2𝑜)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 382   ∨ wo 834   ∨ w3o 1070   ∧ w3a 1071   = wceq 1631   ∈ wcel 2145  ∀wral 3061  ∃wrex 3062  ∅c0 4063  {ctp 4320  ⟨cop 4322   class class class wbr 4786   ↾ cres 5251  Oncon0 5866  ‘cfv 6031  1_𝑜c1o 7706  2_𝑜c2o 7707   No csur 32130   <s cslt 32131

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-ord 5869  df-on 5870  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-1o 7713  df-2o 7714  df-no 32133  df-slt 32134

This theorem is referenced by:  nolesgn2ores  32162