Theorem nolesgn2o 27640

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 1193	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o)) → 𝐵 ∈ No )
2		nofv 27626	. . . . . 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 1488	. . . . 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 1206	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 1o)) → 𝑋 ∈ On)
7		fveq1 6880	. . . . . . . . . . . . 13 ⊢ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) → ((𝐴 ↾ 𝑋)‘𝑦) = ((𝐵 ↾ 𝑋)‘𝑦))
8	7	eqcomd 2742	. . . . . . . . . . . 12 ⊢ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) → ((𝐵 ↾ 𝑋)‘𝑦) = ((𝐴 ↾ 𝑋)‘𝑦))
9	8	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝐵 ↾ 𝑋)‘𝑦) = ((𝐴 ↾ 𝑋)‘𝑦))
10		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ 𝑋)
11	10	fvresd 6901	. . . . . . . . . . 11 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝐵 ↾ 𝑋)‘𝑦) = (𝐵‘𝑦))
12	10	fvresd 6901	. . . . . . . . . . 11 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝐴 ↾ 𝑋)‘𝑦) = (𝐴‘𝑦))
13	9, 11, 12	3eqtr3d 2779	. . . . . . . . . 10 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝑦 ∈ 𝑋) → (𝐵‘𝑦) = (𝐴‘𝑦))
14	13	ralrimiva 3133	. . . . . . . . 9 ⊢ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) → ∀𝑦 ∈ 𝑋 (𝐵‘𝑦) = (𝐴‘𝑦))
15	14	adantr 480	. . . . . . . 8 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) → ∀𝑦 ∈ 𝑋 (𝐵‘𝑦) = (𝐴‘𝑦))
16	15	3ad2ant2 1134	. . . . . . 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 772	. . . . . . . . . . . . 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 550	. . . . . . . . . . 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 550	. . . . . . . . . . 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 966	. . . . . . . . . 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 1117	. . . . . . . . 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 1333	. . . . . . . . . 10 ⊢ (((𝐵‘𝑋) = ∅ ∧ (𝐴‘𝑋) = 2o) → (((𝐵‘𝑋) = 1o ∧ (𝐴‘𝑋) = ∅) ∨ ((𝐵‘𝑋) = 1o ∧ (𝐴‘𝑋) = 2o) ∨ ((𝐵‘𝑋) = ∅ ∧ (𝐴‘𝑋) = 2o)))
25		3mix2 1332	. . . . . . . . . 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 6894	. . . . . . . . 9 ⊢ (𝐵‘𝑋) ∈ V
29		fvex 6894	. . . . . . . . 9 ⊢ (𝐴‘𝑋) ∈ V
30	28, 29	brtp 5503	. . . . . . . 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 234	. . . . . . 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 3306	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (∀𝑦 ∈ 𝑥 (𝐵‘𝑦) = (𝐴‘𝑦) ↔ ∀𝑦 ∈ 𝑋 (𝐵‘𝑦) = (𝐴‘𝑦)))
33		fveq2 6881	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝐵‘𝑥) = (𝐵‘𝑋))
34		fveq2 6881	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝐴‘𝑥) = (𝐴‘𝑋))
35	33, 34	breq12d 5137	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → ((𝐵‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐴‘𝑥) ↔ (𝐵‘𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐴‘𝑋)))
36	32, 35	anbi12d 632	. . . . . . . 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 3606	. . . . . . 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 836	. . . . . 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 1205	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 1o)) → 𝐵 ∈ No )
40		simp11 1204	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 1o)) → 𝐴 ∈ No )
41		sltval 27616	. . . . . . 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 584	. . . . . 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 257	. . . . 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 1121	. . . 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 145	. 2 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o)) → (¬ 𝐵 <s 𝐴 → (𝐵‘𝑋) = 2o))
47	46	3impia 1117	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 206   ∧ wa 395   ∨ wo 847   ∨ w3o 1085   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3052  ∃wrex 3061  ∅c0 4313  {ctp 4610  ⟨cop 4612   class class class wbr 5124   ↾ cres 5661  Oncon0 6357  ‘cfv 6536  1_oc1o 8478  2_oc2o 8479   No csur 27608   <s cslt 27609

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-tp 4611  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ord 6360  df-on 6361  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-1o 8485  df-2o 8486  df-no 27611  df-slt 27612

This theorem is referenced by:  nolesgn2ores  27641