Theorem nogesgn1o 27592

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 1193	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 1o)) → 𝐵 ∈ No )
2		nofv 27576	. . . . . 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 1486	. . . . 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 1206	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 1o) ∧ ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 2o)) → 𝑋 ∈ On)
7		fveq1 6860	. . . . . . . . . . . 12 ⊢ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) → ((𝐴 ↾ 𝑋)‘𝑦) = ((𝐵 ↾ 𝑋)‘𝑦))
8	7	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝐴 ↾ 𝑋)‘𝑦) = ((𝐵 ↾ 𝑋)‘𝑦))
9		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ 𝑋)
10	9	fvresd 6881	. . . . . . . . . . 11 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝐴 ↾ 𝑋)‘𝑦) = (𝐴‘𝑦))
11	9	fvresd 6881	. . . . . . . . . . 11 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝐵 ↾ 𝑋)‘𝑦) = (𝐵‘𝑦))
12	8, 10, 11	3eqtr3d 2773	. . . . . . . . . 10 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝑦 ∈ 𝑋) → (𝐴‘𝑦) = (𝐵‘𝑦))
13	12	ralrimiva 3126	. . . . . . . . 9 ⊢ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) → ∀𝑦 ∈ 𝑋 (𝐴‘𝑦) = (𝐵‘𝑦))
14	13	adantr 480	. . . . . . . 8 ⊢ (((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 1o) → ∀𝑦 ∈ 𝑋 (𝐴‘𝑦) = (𝐵‘𝑦))
15	14	3ad2ant2 1134	. . . . . . 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 1201	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 1o) ∧ ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 2o)) → (𝐴‘𝑋) = 1o)
17		simp3 1138	. . . . . . . . . . 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 511	. . . . . . . . . 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 1009	. . . . . . . . . 10 ⊢ (((𝐴‘𝑋) = 1o ∧ ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 2o)) ↔ (((𝐴‘𝑋) = 1o ∧ (𝐵‘𝑋) = ∅) ∨ ((𝐴‘𝑋) = 1o ∧ (𝐵‘𝑋) = 2o)))
20	18, 19	sylib 218	. . . . . . . . 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 1331	. . . . . . . . . 10 ⊢ (((𝐴‘𝑋) = 1o ∧ (𝐵‘𝑋) = ∅) → (((𝐴‘𝑋) = 1o ∧ (𝐵‘𝑋) = ∅) ∨ ((𝐴‘𝑋) = 1o ∧ (𝐵‘𝑋) = 2o) ∨ ((𝐴‘𝑋) = ∅ ∧ (𝐵‘𝑋) = 2o)))
22		3mix2 1332	. . . . . . . . . 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 857	. . . . . . . . 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 6874	. . . . . . . . 9 ⊢ (𝐴‘𝑋) ∈ V
26		fvex 6874	. . . . . . . . 9 ⊢ (𝐵‘𝑋) ∈ V
27	25, 26	brtp 5486	. . . . . . . 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 234	. . . . . . 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 3298	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (∀𝑦 ∈ 𝑥 (𝐴‘𝑦) = (𝐵‘𝑦) ↔ ∀𝑦 ∈ 𝑋 (𝐴‘𝑦) = (𝐵‘𝑦)))
30		fveq2 6861	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝐴‘𝑥) = (𝐴‘𝑋))
31		fveq2 6861	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝐵‘𝑥) = (𝐵‘𝑋))
32	30, 31	breq12d 5123	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → ((𝐴‘𝑥){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑥) ↔ (𝐴‘𝑋){⟨1o, ∅⟩, ⟨1o, 2o⟩, ⟨∅, 2o⟩} (𝐵‘𝑋)))
33	29, 32	anbi12d 632	. . . . . . . 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 3591	. . . . . . 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 836	. . . . . 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 1204	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 1o) ∧ ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 2o)) → 𝐴 ∈ No )
37		simp12 1205	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 1o) ∧ ((𝐵‘𝑋) = ∅ ∨ (𝐵‘𝑋) = 2o)) → 𝐵 ∈ No )
38		sltval 27566	. . . . . . 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 584	. . . . . 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 257	. . . . 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 1121	. . . 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 1117	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 206   ∧ wa 395   ∨ wo 847   ∨ w3o 1085   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3045  ∃wrex 3054  ∅c0 4299  {ctp 4596  ⟨cop 4598   class class class wbr 5110   ↾ cres 5643  Oncon0 6335  ‘cfv 6514  1_oc1o 8430  2_oc2o 8431   No csur 27558   <s cslt 27559

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 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ord 6338  df-on 6339  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fv 6522  df-1o 8437  df-2o 8438  df-no 27561  df-slt 27562

This theorem is referenced by:  nogesgn1ores  27593