Theorem nogesgn1o 27846

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 1211	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 1o)) → 𝐵 ∈ No )
2		nofv 27830	. . . . . 6 ⊢ (𝐵 ∈ No → ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 1o ∨ (𝐵‘𝑋) = 2o))
3	1, 2	syl 18	. . . . 5 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 1o)) → ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 1o ∨ (𝐵‘𝑋) = 2o))
4		3orel2 1515	. . . . 5 ⊢ (¬ (𝐵‘𝑋) = 1o → (((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 1o ∨ (𝐵‘𝑋) = 2o) → ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 2o)))
5	3, 4	syl5com 32	. . . 4 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 1o)) → (¬ (𝐵‘𝑋) = 1o → ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 2o)))
6		simp13 1224	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 1o) ∧ ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 2o)) → 𝑋 ∈ On)
7		fveq1 6880	. . . . . . . . . . . 12 ⊢ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) → ((𝐴 ↾ 𝑋)‘𝑦) = ((𝐵 ↾ 𝑋)‘𝑦))
8	7	adantr 485	. . . . . . . . . . 11 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝐴 ↾ 𝑋)‘𝑦) = ((𝐵 ↾ 𝑋)‘𝑦))
9		simpr 489	. . . . . . . . . . . 12 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ 𝑋)
10	9	fvresd 6901	. . . . . . . . . . 11 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝐴 ↾ 𝑋)‘𝑦) = (𝐴‘𝑦))
11	9	fvresd 6901	. . . . . . . . . . 11 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝐵 ↾ 𝑋)‘𝑦) = (𝐵‘𝑦))
12	8, 10, 11	3eqtr3d 2806	. . . . . . . . . 10 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝑦 ∈ 𝑋) → (𝐴‘𝑦) = (𝐵‘𝑦))
13	12	ralrimiva 3157	. . . . . . . . 9 ⊢ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) → ∀𝑦 ∈ 𝑋 (𝐴‘𝑦) = (𝐵‘𝑦))
14	13	adantr 485	. . . . . . . 8 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 1o) → ∀𝑦 ∈ 𝑋 (𝐴‘𝑦) = (𝐵‘𝑦))
15	14	3ad2ant2 1152	. . . . . . 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 1219	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 1o) ∧ ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 2o)) → (𝐴‘𝑋) = 1o)
17		simp3 1156	. . . . . . . . . . 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 520	. . . . . . . . . 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 1025	. . . . . . . . . 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 1349	. . . . . . . . . 10 ⊢ (((𝐴‘𝑋) = 1o ∧ (𝐵‘𝑋) = ∅) → (((𝐴‘𝑋) = 1o ∧ (𝐵‘𝑋) = ∅) ∨ ((𝐴‘𝑋) = 1o ∧ (𝐵‘𝑋) = 2o) ∨ ((𝐴‘𝑋) = ∅ ∧ (𝐵‘𝑋) = 2o)))
22		3mix2 1350	. . . . . . . . . 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 870	. . . . . . . . 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 18	. . . . . . . 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 6894	. . . . . . . . 9 ⊢ (𝐴‘𝑋) ∈ V
26		fvex 6894	. . . . . . . . 9 ⊢ (𝐵‘𝑋) ∈ V
27	25, 26	brtp 5507	. . . . . . . 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 3320	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦) ↔ ∀𝑦 ∈ 𝑋 (𝐴‘𝑦) = (𝐵‘𝑦)))
30		fveq2 6881	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝐴‘𝑥) = (𝐴‘𝑋))
31		fveq2 6881	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝐵‘𝑥) = (𝐵‘𝑋))
32	30, 31	breq12d 5122	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → ((𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑥) ↔ (𝐴‘𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑋)))
33	29, 32	anbi12d 643	. . . . . . . 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 3581	. . . . . . 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 849	. . . . . 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 1222	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 1o) ∧ ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 2o)) → 𝐴 ∈ No )
37		simp12 1223	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 1o) ∧ ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 2o)) → 𝐵 ∈ No )
38		ltsval 27820	. . . . . . 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 595	. . . . . 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 1139	. . . 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 48	. . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 1o)) → (¬ (𝐵‘𝑋) = 1o → 𝐴 <s 𝐵))
43	42	con1d 146	. 2 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 1o)) → (¬ 𝐴 <s 𝐵 → (𝐵‘𝑋) = 1o))
44	43	3impia 1135	1 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) → (𝐵‘𝑋) = 1o)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∨ w3o 1102   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089  ∅c0 4286  {ctp 4593  ⟨cop 4595   class class class wbr 5109   ↾ cres 5663  Oncon0 6360  ‘cfv 6536  1_oc1o 8442  2_oc2o 8443   No csur 27813   <s clts 27814

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ord 6363  df-on 6364  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-1o 8449  df-2o 8450  df-no 27816  df-lts 27817

This theorem is used by:  nogesgn1ores  27847