Theorem nolesgn2o 32152

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 1230	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2𝑜)) → 𝐵 ∈ No )
2		nofv 32138	. . . . . 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 31922	. . . . 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 1248	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2𝑜) ∧ ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 1𝑜)) → 𝑋 ∈ On)
7		fveq1 6353	. . . . . . . . . . . . 13 ⊢ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) → ((𝐴 ↾ 𝑋)‘𝑦) = ((𝐵 ↾ 𝑋)‘𝑦))
8	7	eqcomd 2767	. . . . . . . . . . . 12 ⊢ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) → ((𝐵 ↾ 𝑋)‘𝑦) = ((𝐴 ↾ 𝑋)‘𝑦))
9	8	adantr 472	. . . . . . . . . . 11 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝐵 ↾ 𝑋)‘𝑦) = ((𝐴 ↾ 𝑋)‘𝑦))
10		simpr 479	. . . . . . . . . . . 12 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ 𝑋)
11	10	fvresd 6371	. . . . . . . . . . 11 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝐵 ↾ 𝑋)‘𝑦) = (𝐵‘𝑦))
12	10	fvresd 6371	. . . . . . . . . . 11 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝐴 ↾ 𝑋)‘𝑦) = (𝐴‘𝑦))
13	9, 11, 12	3eqtr3d 2803	. . . . . . . . . 10 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝑦 ∈ 𝑋) → (𝐵‘𝑦) = (𝐴‘𝑦))
14	13	ralrimiva 3105	. . . . . . . . 9 ⊢ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) → ∀𝑦 ∈ 𝑋 (𝐵‘𝑦) = (𝐴‘𝑦))
15	14	adantr 472	. . . . . . . 8 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2𝑜) → ∀𝑦 ∈ 𝑋 (𝐵‘𝑦) = (𝐴‘𝑦))
16	15	3ad2ant2 1129	. . . . . . 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 813	. . . . . . . . . . . . 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 577	. . . . . . . . . . 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 577	. . . . . . . . . . 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 919	. . . . . . . . . 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 1110	. . . . . . . . 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 1417	. . . . . . . . . 10 ⊢ (((𝐵‘𝑋) = ∅ ∧ (𝐴‘𝑋) = 2𝑜) → (((𝐵‘𝑋) = 1𝑜 ∧ (𝐴‘𝑋) = ∅) ∨ ((𝐵‘𝑋) = 1𝑜 ∧ (𝐴‘𝑋) = 2𝑜) ∨ ((𝐵‘𝑋) = ∅ ∧ (𝐴‘𝑋) = 2𝑜)))
25		3mix2 1416	. . . . . . . . . 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 393	. . . . . . . . 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 6364	. . . . . . . . 9 ⊢ (𝐵‘𝑋) ∈ V
29		fvex 6364	. . . . . . . . 9 ⊢ (𝐴‘𝑋) ∈ V
30	28, 29	brtp 31968	. . . . . . . 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 3278	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (∀𝑦 ∈ 𝑥 (𝐵‘𝑦) = (𝐴‘𝑦) ↔ ∀𝑦 ∈ 𝑋 (𝐵‘𝑦) = (𝐴‘𝑦)))
33		fveq2 6354	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝐵‘𝑥) = (𝐵‘𝑋))
34		fveq2 6354	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝐴‘𝑥) = (𝐴‘𝑋))
35	33, 34	breq12d 4818	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → ((𝐵‘𝑥){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐴‘𝑥) ↔ (𝐵‘𝑋){⟨1𝑜, ∅⟩, ⟨1𝑜, 2𝑜⟩, ⟨∅, 2𝑜⟩} (𝐴‘𝑋)))
36	32, 35	anbi12d 749	. . . . . . . 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 3450	. . . . . . 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 1475	. . . . . 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 1247	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2𝑜) ∧ ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 1𝑜)) → 𝐵 ∈ No )
40		simp11 1246	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2𝑜) ∧ ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 1𝑜)) → 𝐴 ∈ No )
41		sltval 32128	. . . . . . 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 696	. . . . . 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 1115	. . . 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 1110	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 382   ∧ wa 383   ∨ w3o 1071   ∧ w3a 1072   = wceq 1632   ∈ wcel 2140  ∀wral 3051  ∃wrex 3052  ∅c0 4059  {ctp 4326  ⟨cop 4328   class class class wbr 4805   ↾ cres 5269  Oncon0 5885  ‘cfv 6050  1_𝑜c1o 7724  2_𝑜c2o 7725   No csur 32121   <s cslt 32122

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-pss 3732  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-tp 4327  df-op 4329  df-uni 4590  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-tr 4906  df-id 5175  df-eprel 5180  df-po 5188  df-so 5189  df-fr 5226  df-we 5228  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-ord 5888  df-on 5889  df-suc 5891  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-1o 7731  df-2o 7732  df-no 32124  df-slt 32125

This theorem is referenced by:  nolesgn2ores  32153