Theorem nolesgn2o 27163

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 1192	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o)) → 𝐵 ∈ No )
2		nofv 27149	. . . . . 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 1486	. . . . 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 1205	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 1o)) → 𝑋 ∈ On)
7		fveq1 6887	. . . . . . . . . . . . 13 ⊢ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) → ((𝐴 ↾ 𝑋)‘𝑦) = ((𝐵 ↾ 𝑋)‘𝑦))
8	7	eqcomd 2738	. . . . . . . . . . . 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 485	. . . . . . . . . . . 12 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ 𝑋)
11	10	fvresd 6908	. . . . . . . . . . 11 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝐵 ↾ 𝑋)‘𝑦) = (𝐵‘𝑦))
12	10	fvresd 6908	. . . . . . . . . . 11 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝐴 ↾ 𝑋)‘𝑦) = (𝐴‘𝑦))
13	9, 11, 12	3eqtr3d 2780	. . . . . . . . . 10 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝑦 ∈ 𝑋) → (𝐵‘𝑦) = (𝐴‘𝑦))
14	13	ralrimiva 3146	. . . . . . . . 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 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 771	. . . . . . . . . . . . 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 551	. . . . . . . . . . 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 551	. . . . . . . . . . 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 963	. . . . . . . . . 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 1332	. . . . . . . . . 10 ⊢ (((𝐵‘𝑋) = ∅ ∧ (𝐴‘𝑋) = 2o) → (((𝐵‘𝑋) = 1o ∧ (𝐴‘𝑋) = ∅) ∨ ((𝐵‘𝑋) = 1o ∧ (𝐴‘𝑋) = 2o) ∨ ((𝐵‘𝑋) = ∅ ∧ (𝐴‘𝑋) = 2o)))
25		3mix2 1331	. . . . . . . . . 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 855	. . . . . . . . 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 6901	. . . . . . . . 9 ⊢ (𝐵‘𝑋) ∈ V
29		fvex 6901	. . . . . . . . 9 ⊢ (𝐴‘𝑋) ∈ V
30	28, 29	brtp 5522	. . . . . . . 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 233	. . . . . . 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 3322	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (∀𝑦 ∈ 𝑥 (𝐵‘𝑦) = (𝐴‘𝑦) ↔ ∀𝑦 ∈ 𝑋 (𝐵‘𝑦) = (𝐴‘𝑦)))
33		fveq2 6888	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝐵‘𝑥) = (𝐵‘𝑋))
34		fveq2 6888	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝐴‘𝑥) = (𝐴‘𝑋))
35	33, 34	breq12d 5160	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → ((𝐵‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐴‘𝑥) ↔ (𝐵‘𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐴‘𝑋)))
36	32, 35	anbi12d 631	. . . . . . . 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 3612	. . . . . . 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 835	. . . . . 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 1204	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 1o)) → 𝐵 ∈ No )
40		simp11 1203	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2o) ∧ ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 1o)) → 𝐴 ∈ No )
41		sltval 27139	. . . . . . 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 256	. . . . 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 205   ∧ wa 396   ∨ wo 845   ∨ w3o 1086   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3061  ∃wrex 3070  ∅c0 4321  {ctp 4631  ⟨cop 4633   class class class wbr 5147   ↾ cres 5677  Oncon0 6361  ‘cfv 6540  1_oc1o 8455  2_oc2o 8456   No csur 27132   <s cslt 27133

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6364  df-on 6365  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-1o 8462  df-2o 8463  df-no 27135  df-slt 27136

This theorem is referenced by:  nolesgn2ores  27164