Theorem nogesgn1o 33441

Description: Given 𝐴 greater than or equal to 𝐵, equal to 𝐵 up to 𝑋, and 𝐴(𝑋) = 1_o, then 𝐵(𝑋) = 1_o. (Contributed by Scott Fenton, 9-Aug-2024.)

Assertion

Ref	Expression
nogesgn1o	⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) → (𝐵‘𝑋) = 1o)

Proof of Theorem nogesgn1o

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl2 1189	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 1o)) → 𝐵 ∈ No )
2		nofv 33425	. . . . . 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) = 1o)) → ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 1o ∨ (𝐵‘𝑋) = 2o))
4		3orel2 33172	. . . . 5 ⊢ (¬ (𝐵‘𝑋) = 1o → (((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 1o ∨ (𝐵‘𝑋) = 2o) → ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 2o)))
5	3, 4	syl5com 31	. . . 4 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 1o)) → (¬ (𝐵‘𝑋) = 1o → ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 2o)))
6		simp13 1202	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 1o) ∧ ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 2o)) → 𝑋 ∈ On)
7		fveq1 6657	. . . . . . . . . . . 12 ⊢ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) → ((𝐴 ↾ 𝑋)‘𝑦) = ((𝐵 ↾ 𝑋)‘𝑦))
8	7	adantr 484	. . . . . . . . . . 11 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝐴 ↾ 𝑋)‘𝑦) = ((𝐵 ↾ 𝑋)‘𝑦))
9		simpr 488	. . . . . . . . . . . 12 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ 𝑋)
10	9	fvresd 6678	. . . . . . . . . . 11 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝐴 ↾ 𝑋)‘𝑦) = (𝐴‘𝑦))
11	9	fvresd 6678	. . . . . . . . . . 11 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝐵 ↾ 𝑋)‘𝑦) = (𝐵‘𝑦))
12	8, 10, 11	3eqtr3d 2801	. . . . . . . . . 10 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝑦 ∈ 𝑋) → (𝐴‘𝑦) = (𝐵‘𝑦))
13	12	ralrimiva 3113	. . . . . . . . 9 ⊢ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) → ∀𝑦 ∈ 𝑋 (𝐴‘𝑦) = (𝐵‘𝑦))
14	13	adantr 484	. . . . . . . 8 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 1o) → ∀𝑦 ∈ 𝑋 (𝐴‘𝑦) = (𝐵‘𝑦))
15	14	3ad2ant2 1131	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 1o) ∧ ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 2o)) → ∀𝑦 ∈ 𝑋 (𝐴‘𝑦) = (𝐵‘𝑦))
16		simp2r 1197	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 1o) ∧ ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 2o)) → (𝐴‘𝑋) = 1o)
17		simp3 1135	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 1o) ∧ ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 2o)) → ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 2o))
18	16, 17	jca 515	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 1o) ∧ ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 2o)) → ((𝐴‘𝑋) = 1o ∧ ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 2o)))
19		andi 1005	. . . . . . . . . 10 ⊢ (((𝐴‘𝑋) = 1o ∧ ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 2o)) ↔ (((𝐴‘𝑋) = 1o ∧ (𝐵‘𝑋) = ∅) ∨ ((𝐴‘𝑋) = 1o ∧ (𝐵‘𝑋) = 2o)))
20	18, 19	sylib 221	. . . . . . . . 9 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 1o) ∧ ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 2o)) → (((𝐴‘𝑋) = 1o ∧ (𝐵‘𝑋) = ∅) ∨ ((𝐴‘𝑋) = 1o ∧ (𝐵‘𝑋) = 2o)))
21		3mix1 1327	. . . . . . . . . 10 ⊢ (((𝐴‘𝑋) = 1o ∧ (𝐵‘𝑋) = ∅) → (((𝐴‘𝑋) = 1o ∧ (𝐵‘𝑋) = ∅) ∨ ((𝐴‘𝑋) = 1o ∧ (𝐵‘𝑋) = 2o) ∨ ((𝐴‘𝑋) = ∅ ∧ (𝐵‘𝑋) = 2o)))
22		3mix2 1328	. . . . . . . . . 10 ⊢ (((𝐴‘𝑋) = 1o ∧ (𝐵‘𝑋) = 2o) → (((𝐴‘𝑋) = 1o ∧ (𝐵‘𝑋) = ∅) ∨ ((𝐴‘𝑋) = 1o ∧ (𝐵‘𝑋) = 2o) ∨ ((𝐴‘𝑋) = ∅ ∧ (𝐵‘𝑋) = 2o)))
23	21, 22	jaoi 854	. . . . . . . . 9 ⊢ ((((𝐴‘𝑋) = 1o ∧ (𝐵‘𝑋) = ∅) ∨ ((𝐴‘𝑋) = 1o ∧ (𝐵‘𝑋) = 2o)) → (((𝐴‘𝑋) = 1o ∧ (𝐵‘𝑋) = ∅) ∨ ((𝐴‘𝑋) = 1o ∧ (𝐵‘𝑋) = 2o) ∨ ((𝐴‘𝑋) = ∅ ∧ (𝐵‘𝑋) = 2o)))
24	20, 23	syl 17	. . . . . . . 8 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 1o) ∧ ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 2o)) → (((𝐴‘𝑋) = 1o ∧ (𝐵‘𝑋) = ∅) ∨ ((𝐴‘𝑋) = 1o ∧ (𝐵‘𝑋) = 2o) ∨ ((𝐴‘𝑋) = ∅ ∧ (𝐵‘𝑋) = 2o)))
25		fvex 6671	. . . . . . . . 9 ⊢ (𝐴‘𝑋) ∈ V
26		fvex 6671	. . . . . . . . 9 ⊢ (𝐵‘𝑋) ∈ V
27	25, 26	brtp 33232	. . . . . . . 8 ⊢ ((𝐴‘𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑋) ↔ (((𝐴‘𝑋) = 1o ∧ (𝐵‘𝑋) = ∅) ∨ ((𝐴‘𝑋) = 1o ∧ (𝐵‘𝑋) = 2o) ∨ ((𝐴‘𝑋) = ∅ ∧ (𝐵‘𝑋) = 2o)))
28	24, 27	sylibr 237	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 1o) ∧ ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 2o)) → (𝐴‘𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑋))
29		raleq 3323	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦) ↔ ∀𝑦 ∈ 𝑋 (𝐴‘𝑦) = (𝐵‘𝑦)))
30		fveq2 6658	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝐴‘𝑥) = (𝐴‘𝑋))
31		fveq2 6658	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝐵‘𝑥) = (𝐵‘𝑋))
32	30, 31	breq12d 5045	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → ((𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑥) ↔ (𝐴‘𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑋)))
33	29, 32	anbi12d 633	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ((∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦) ∧ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑥)) ↔ (∀𝑦 ∈ 𝑋 (𝐴‘𝑦) = (𝐵‘𝑦) ∧ (𝐴‘𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑋))))
34	33	rspcev 3541	. . . . . . 7 ⊢ ((𝑋 ∈ On ∧ (∀𝑦 ∈ 𝑋 (𝐴‘𝑦) = (𝐵‘𝑦) ∧ (𝐴‘𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑋))) → ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦) ∧ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑥)))
35	6, 15, 28, 34	syl12anc 835	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 1o) ∧ ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 2o)) → ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦) ∧ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑥)))
36		simp11 1200	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 1o) ∧ ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 2o)) → 𝐴 ∈ No )
37		simp12 1201	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 1o) ∧ ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 2o)) → 𝐵 ∈ No )
38		sltval 33415	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 <s 𝐵 ↔ ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦) ∧ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑥))))
39	36, 37, 38	syl2anc 587	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 1o) ∧ ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 2o)) → (𝐴 <s 𝐵 ↔ ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦) ∧ (𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑥))))
40	35, 39	mpbird 260	. . . . 5 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 1o) ∧ ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 2o)) → 𝐴 <s 𝐵)
41	40	3expia 1118	. . . 4 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 1o)) → (((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 2o) → 𝐴 <s 𝐵))
42	5, 41	syld 47	. . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 1o)) → (¬ (𝐵‘𝑋) = 1o → 𝐴 <s 𝐵))
43	42	con1d 147	. 2 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 1o)) → (¬ 𝐴 <s 𝐵 → (𝐵‘𝑋) = 1o))
44	43	3impia 1114	1 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) → (𝐵‘𝑋) = 1o)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∨ w3o 1083   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3070  ∃wrex 3071  ∅c0 4225  {ctp 4526  ⟨cop 4528   class class class wbr 5032   ↾ cres 5526  Oncon0 6169  ‘cfv 6335  1_oc1o 8105  2_oc2o 8106   No csur 33408   <s cslt 33409

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pr 5298  ax-un 7459

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-ord 6172  df-on 6173  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-1o 8112  df-2o 8113  df-no 33411  df-slt 33412

This theorem is referenced by:  nogesgn1ores  33442