Theorem nogesgn1o 27739

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 1207	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 1o)) → 𝐵 ∈ No )
2		nofv 27723	. . . . . 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 1507	. . . . 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 1220	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 1o) ∧ ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 2o)) → 𝑋 ∈ On)
7		fveq1 6868	. . . . . . . . . . . 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 6889	. . . . . . . . . . 11 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝐴 ↾ 𝑋)‘𝑦) = (𝐴‘𝑦))
11	9	fvresd 6889	. . . . . . . . . . 11 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝐵 ↾ 𝑋)‘𝑦) = (𝐵‘𝑦))
12	8, 10, 11	3eqtr3d 2807	. . . . . . . . . 10 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝑦 ∈ 𝑋) → (𝐴‘𝑦) = (𝐵‘𝑦))
13	12	ralrimiva 3156	. . . . . . . . 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 1148	. . . . . . 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 1215	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 1o) ∧ ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 2o)) → (𝐴‘𝑋) = 1o)
17		simp3 1152	. . . . . . . . . . 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 519	. . . . . . . . . 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 1021	. . . . . . . . . 10 ⊢ (((𝐴‘𝑋) = 1o ∧ ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 2o)) ↔ (((𝐴‘𝑋) = 1o ∧ (𝐵‘𝑋) = ∅) ∨ ((𝐴‘𝑋) = 1o ∧ (𝐵‘𝑋) = 2o)))
20	18, 19	sylib 220	. . . . . . . . 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 1345	. . . . . . . . . 10 ⊢ (((𝐴‘𝑋) = 1o ∧ (𝐵‘𝑋) = ∅) → (((𝐴‘𝑋) = 1o ∧ (𝐵‘𝑋) = ∅) ∨ ((𝐴‘𝑋) = 1o ∧ (𝐵‘𝑋) = 2o) ∨ ((𝐴‘𝑋) = ∅ ∧ (𝐵‘𝑋) = 2o)))
22		3mix2 1346	. . . . . . . . . 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 868	. . . . . . . . 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 6882	. . . . . . . . 9 ⊢ (𝐴‘𝑋) ∈ V
26		fvex 6882	. . . . . . . . 9 ⊢ (𝐵‘𝑋) ∈ V
27	25, 26	brtp 5495	. . . . . . . 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 236	. . . . . . 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 3319	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦) ↔ ∀𝑦 ∈ 𝑋 (𝐴‘𝑦) = (𝐵‘𝑦)))
30		fveq2 6869	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝐴‘𝑥) = (𝐴‘𝑋))
31		fveq2 6869	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝐵‘𝑥) = (𝐵‘𝑋))
32	30, 31	breq12d 5115	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → ((𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑥) ↔ (𝐴‘𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑋)))
33	29, 32	anbi12d 641	. . . . . . . 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 3583	. . . . . . 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 847	. . . . . 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 1218	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 1o) ∧ ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 2o)) → 𝐴 ∈ No )
37		simp12 1219	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 1o) ∧ ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 2o)) → 𝐵 ∈ No )
38		ltsval 27713	. . . . . . 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 593	. . . . . 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 259	. . . . 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 1135	. . . 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 145	. 2 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 1o)) → (¬ 𝐴 <s 𝐵 → (𝐵‘𝑋) = 1o))
44	43	3impia 1131	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 208   ∧ wa 399   ∨ wo 858   ∨ w3o 1098   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144  ∀wral 3078  ∃wrex 3088  ∅c0 4287  {ctp 4588  ⟨cop 4590   class class class wbr 5102   ↾ cres 5651  Oncon0 6348  ‘cfv 6523  1_oc1o 8432  2_oc2o 8433   No csur 27706   <s clts 27707

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-ne 2960  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ord 6351  df-on 6352  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-fv 6531  df-1o 8439  df-2o 8440  df-no 27709  df-lts 27710

This theorem is referenced by:  nogesgn1ores  27740