Theorem nolesgn2o 33568

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 1194	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o)) → 𝐵 ∈ No )
2		nofv 33554	. . . . . 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 33342	. . . . 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 1207	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 1o)) → 𝑋 ∈ On)
7		fveq1 6705	. . . . . . . . . . . . 13 ⊢ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) → ((𝐴 ↾ 𝑋)‘𝑦) = ((𝐵 ↾ 𝑋)‘𝑦))
8	7	eqcomd 2740	. . . . . . . . . . . 12 ⊢ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) → ((𝐵 ↾ 𝑋)‘𝑦) = ((𝐴 ↾ 𝑋)‘𝑦))
9	8	adantr 484	. . . . . . . . . . 11 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝐵 ↾ 𝑋)‘𝑦) = ((𝐴 ↾ 𝑋)‘𝑦))
10		simpr 488	. . . . . . . . . . . 12 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ 𝑋)
11	10	fvresd 6726	. . . . . . . . . . 11 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝐵 ↾ 𝑋)‘𝑦) = (𝐵‘𝑦))
12	10	fvresd 6726	. . . . . . . . . . 11 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝐴 ↾ 𝑋)‘𝑦) = (𝐴‘𝑦))
13	9, 11, 12	3eqtr3d 2782	. . . . . . . . . 10 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝑦 ∈ 𝑋) → (𝐵‘𝑦) = (𝐴‘𝑦))
14	13	ralrimiva 3098	. . . . . . . . 9 ⊢ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) → ∀𝑦 ∈ 𝑋 (𝐵‘𝑦) = (𝐴‘𝑦))
15	14	adantr 484	. . . . . . . 8 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) → ∀𝑦 ∈ 𝑋 (𝐵‘𝑦) = (𝐴‘𝑦))
16	15	3ad2ant2 1136	. . . . . . 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 773	. . . . . . . . . . . . 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 554	. . . . . . . . . . 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 554	. . . . . . . . . . 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 965	. . . . . . . . . 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 1119	. . . . . . . . 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 1334	. . . . . . . . . 10 ⊢ (((𝐵‘𝑋) = ∅ ∧ (𝐴‘𝑋) = 2o) → (((𝐵‘𝑋) = 1o ∧ (𝐴‘𝑋) = ∅) ∨ ((𝐵‘𝑋) = 1o ∧ (𝐴‘𝑋) = 2o) ∨ ((𝐵‘𝑋) = ∅ ∧ (𝐴‘𝑋) = 2o)))
25		3mix2 1333	. . . . . . . . . 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 857	. . . . . . . . 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 6719	. . . . . . . . 9 ⊢ (𝐵‘𝑋) ∈ V
29		fvex 6719	. . . . . . . . 9 ⊢ (𝐴‘𝑋) ∈ V
30	28, 29	brtp 33404	. . . . . . . 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 237	. . . . . . 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 3312	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (∀𝑦 ∈ 𝑥 (𝐵‘𝑦) = (𝐴‘𝑦) ↔ ∀𝑦 ∈ 𝑋 (𝐵‘𝑦) = (𝐴‘𝑦)))
33		fveq2 6706	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝐵‘𝑥) = (𝐵‘𝑋))
34		fveq2 6706	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝐴‘𝑥) = (𝐴‘𝑋))
35	33, 34	breq12d 5056	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → ((𝐵‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐴‘𝑥) ↔ (𝐵‘𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐴‘𝑋)))
36	32, 35	anbi12d 634	. . . . . . . 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 3530	. . . . . . 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 837	. . . . . 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 1206	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 1o)) → 𝐵 ∈ No )
40		simp11 1205	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 1o)) → 𝐴 ∈ No )
41		sltval 33544	. . . . . . 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 587	. . . . . 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 260	. . . . 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 1123	. . . 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 147	. 2 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o)) → (¬ 𝐵 <s 𝐴 → (𝐵‘𝑋) = 2o))
47	46	3impia 1119	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 209   ∧ wa 399   ∨ wo 847   ∨ w3o 1088   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110  ∀wral 3054  ∃wrex 3055  ∅c0 4227  {ctp 4535  ⟨cop 4537   class class class wbr 5043   ↾ cres 5542  Oncon0 6202  ‘cfv 6369  1_oc1o 8184  2_oc2o 8185   No csur 33537   <s cslt 33538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2706  ax-rep 5168  ax-sep 5181  ax-nul 5188  ax-pr 5311  ax-un 7512

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2537  df-eu 2566  df-clab 2713  df-cleq 2726  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3403  df-sbc 3688  df-csb 3803  df-dif 3860  df-un 3862  df-in 3864  df-ss 3874  df-pss 3876  df-nul 4228  df-if 4430  df-pw 4505  df-sn 4532  df-pr 4534  df-tp 4536  df-op 4538  df-uni 4810  df-iun 4896  df-br 5044  df-opab 5106  df-mpt 5125  df-tr 5151  df-id 5444  df-eprel 5449  df-po 5457  df-so 5458  df-fr 5498  df-we 5500  df-xp 5546  df-rel 5547  df-cnv 5548  df-co 5549  df-dm 5550  df-rn 5551  df-res 5552  df-ima 5553  df-ord 6205  df-on 6206  df-suc 6208  df-iota 6327  df-fun 6371  df-fn 6372  df-f 6373  df-f1 6374  df-fo 6375  df-f1o 6376  df-fv 6377  df-1o 8191  df-2o 8192  df-no 33540  df-slt 33541

This theorem is referenced by:  nolesgn2ores  33569