Theorem nolesgn2o 31824

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 1065	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2𝑜)) → 𝐵 ∈ No )
2		nofv 31810	. . . . . 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 31593	. . . . 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 1093	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2𝑜) ∧ ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 1𝑜)) → 𝑋 ∈ On)
7		fveq1 6190	. . . . . . . . . . . . 13 ⊢ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) → ((𝐴 ↾ 𝑋)‘𝑦) = ((𝐵 ↾ 𝑋)‘𝑦))
8	7	eqcomd 2628	. . . . . . . . . . . 12 ⊢ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) → ((𝐵 ↾ 𝑋)‘𝑦) = ((𝐴 ↾ 𝑋)‘𝑦))
9	8	adantr 481	. . . . . . . . . . 11 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝐵 ↾ 𝑋)‘𝑦) = ((𝐴 ↾ 𝑋)‘𝑦))
10		simpr 477	. . . . . . . . . . . 12 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ 𝑋)
11	10	fvresd 6208	. . . . . . . . . . 11 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝐵 ↾ 𝑋)‘𝑦) = (𝐵‘𝑦))
12	10	fvresd 6208	. . . . . . . . . . 11 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝐴 ↾ 𝑋)‘𝑦) = (𝐴‘𝑦))
13	9, 11, 12	3eqtr3d 2664	. . . . . . . . . 10 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝑦 ∈ 𝑋) → (𝐵‘𝑦) = (𝐴‘𝑦))
14	13	ralrimiva 2966	. . . . . . . . 9 ⊢ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) → ∀𝑦 ∈ 𝑋 (𝐵‘𝑦) = (𝐴‘𝑦))
15	14	adantr 481	. . . . . . . 8 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2𝑜) → ∀𝑦 ∈ 𝑋 (𝐵‘𝑦) = (𝐴‘𝑦))
16	15	3ad2ant2 1083	. . . . . . 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 796	. . . . . . . . . . . . 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 576	. . . . . . . . . . 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 576	. . . . . . . . . . 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 883	. . . . . . . . . 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 1261	. . . . . . . . 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 1232	. . . . . . . . . 10 ⊢ (((𝐵‘𝑋) = ∅ ∧ (𝐴‘𝑋) = 2𝑜) → (((𝐵‘𝑋) = 1𝑜 ∧ (𝐴‘𝑋) = ∅) ∨ ((𝐵‘𝑋) = 1𝑜 ∧ (𝐴‘𝑋) = 2𝑜) ∨ ((𝐵‘𝑋) = ∅ ∧ (𝐴‘𝑋) = 2𝑜)))
25		3mix2 1231	. . . . . . . . . 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 394	. . . . . . . . 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 6201	. . . . . . . . 9 ⊢ (𝐵‘𝑋) ∈ V
29		fvex 6201	. . . . . . . . 9 ⊢ (𝐴‘𝑋) ∈ V
30	28, 29	brtp 31639	. . . . . . . 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 3138	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (∀𝑦 ∈ 𝑥 (𝐵‘𝑦) = (𝐴‘𝑦) ↔ ∀𝑦 ∈ 𝑋 (𝐵‘𝑦) = (𝐴‘𝑦)))
33		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝐵‘𝑥) = (𝐵‘𝑋))
34		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝐴‘𝑥) = (𝐴‘𝑋))
35	33, 34	breq12d 4666	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → ((𝐵‘𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐴‘𝑥) ↔ (𝐵‘𝑋){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐴‘𝑋)))
36	32, 35	anbi12d 747	. . . . . . . 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 3309	. . . . . . 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 1324	. . . . . 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 1092	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2𝑜) ∧ ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 1𝑜)) → 𝐵 ∈ No )
40		simp11 1091	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2𝑜) ∧ ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 1𝑜)) → 𝐴 ∈ No )
41		sltval 31800	. . . . . . 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 693	. . . . . 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 1267	. . . 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 139	. 2 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2𝑜)) → (¬ 𝐵 <s 𝐴 → (𝐵‘𝑋) = 2𝑜))
47	46	3impia 1261	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   ∨ wo 383   ∧ wa 384   ∨ w3o 1036   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  ∅c0 3915  {ctp 4181  ⟨cop 4183   class class class wbr 4653   ↾ cres 5116  Oncon0 5723  ‘cfv 5888  1_𝑜c1o 7553  2_𝑜c2o 7554   No csur 31793   <s cslt 31794

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-1o 7560  df-2o 7561  df-no 31796  df-slt 31797

This theorem is referenced by:  nolesgn2ores  31825