Theorem oldfib 28387

Description: The old set of an ordinal is finite iff the ordinal is finite. (Contributed by Scott Fenton, 19-Feb-2026.)

Assertion

Ref	Expression
oldfib	⊢ (𝐴 ∈ On → (𝐴 ∈ ω ↔ ( O ‘𝐴) ∈ Fin))

Proof of Theorem oldfib

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

Step	Hyp	Ref	Expression
1		oldfi 27924	. 2 ⊢ (𝐴 ∈ ω → ( O ‘𝐴) ∈ Fin)
2		fveq2 6827	. . . . 5 ⊢ (𝑥 = 𝑦 → ( O ‘𝑥) = ( O ‘𝑦))
3	2	eleq1d 2824	. . . 4 ⊢ (𝑥 = 𝑦 → (( O ‘𝑥) ∈ Fin ↔ ( O ‘𝑦) ∈ Fin))
4		eleq1 2827	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ ω ↔ 𝑦 ∈ ω))
5	3, 4	imbi12d 345	. . 3 ⊢ (𝑥 = 𝑦 → ((( O ‘𝑥) ∈ Fin → 𝑥 ∈ ω) ↔ (( O ‘𝑦) ∈ Fin → 𝑦 ∈ ω)))
6		fveq2 6827	. . . . 5 ⊢ (𝑥 = 𝐴 → ( O ‘𝑥) = ( O ‘𝐴))
7	6	eleq1d 2824	. . . 4 ⊢ (𝑥 = 𝐴 → (( O ‘𝑥) ∈ Fin ↔ ( O ‘𝐴) ∈ Fin))
8		eleq1 2827	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ ω ↔ 𝐴 ∈ ω))
9	7, 8	imbi12d 345	. . 3 ⊢ (𝑥 = 𝐴 → ((( O ‘𝑥) ∈ Fin → 𝑥 ∈ ω) ↔ (( O ‘𝐴) ∈ Fin → 𝐴 ∈ ω)))
10		oldval 27844	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → ( O ‘𝑥) = ∪ ( M “ 𝑥))
11	10	eleq1d 2824	. . . . . . . . . 10 ⊢ (𝑥 ∈ On → (( O ‘𝑥) ∈ Fin ↔ ∪ ( M “ 𝑥) ∈ Fin))
12	11	biimpa 477	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ ( O ‘𝑥) ∈ Fin) → ∪ ( M “ 𝑥) ∈ Fin)
13		unifi3 9262	. . . . . . . . 9 ⊢ (∪ ( M “ 𝑥) ∈ Fin → ( M “ 𝑥) ⊆ Fin)
14	12, 13	syl 17	. . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ ( O ‘𝑥) ∈ Fin) → ( M “ 𝑥) ⊆ Fin)
15		madef 27846	. . . . . . . . . . 11 ⊢ M :On⟶𝒫 No
16		ffun 6658	. . . . . . . . . . 11 ⊢ ( M :On⟶𝒫 No → Fun M )
17	15, 16	ax-mp 5	. . . . . . . . . 10 ⊢ Fun M
18		onss 7728	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → 𝑥 ⊆ On)
19	15	fdmi 6666	. . . . . . . . . . 11 ⊢ dom M = On
20	18, 19	sseqtrrdi 3956	. . . . . . . . . 10 ⊢ (𝑥 ∈ On → 𝑥 ⊆ dom M )
21		funimass4 6891	. . . . . . . . . 10 ⊢ ((Fun M ∧ 𝑥 ⊆ dom M ) → (( M “ 𝑥) ⊆ Fin ↔ ∀𝑦 ∈ 𝑥 ( M ‘𝑦) ∈ Fin))
22	17, 20, 21	sylancr 593	. . . . . . . . 9 ⊢ (𝑥 ∈ On → (( M “ 𝑥) ⊆ Fin ↔ ∀𝑦 ∈ 𝑥 ( M ‘𝑦) ∈ Fin))
23	22	adantr 481	. . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ ( O ‘𝑥) ∈ Fin) → (( M “ 𝑥) ⊆ Fin ↔ ∀𝑦 ∈ 𝑥 ( M ‘𝑦) ∈ Fin))
24	14, 23	mpbid 233	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ ( O ‘𝑥) ∈ Fin) → ∀𝑦 ∈ 𝑥 ( M ‘𝑦) ∈ Fin)
25		oldssmade 27877	. . . . . . . . 9 ⊢ ( O ‘𝑦) ⊆ ( M ‘𝑦)
26		ssfi 9097	. . . . . . . . 9 ⊢ ((( M ‘𝑦) ∈ Fin ∧ ( O ‘𝑦) ⊆ ( M ‘𝑦)) → ( O ‘𝑦) ∈ Fin)
27	25, 26	mpan2 697	. . . . . . . 8 ⊢ (( M ‘𝑦) ∈ Fin → ( O ‘𝑦) ∈ Fin)
28	27	ralimi 3076	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝑥 ( M ‘𝑦) ∈ Fin → ∀𝑦 ∈ 𝑥 ( O ‘𝑦) ∈ Fin)
29	24, 28	syl 17	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ ( O ‘𝑥) ∈ Fin) → ∀𝑦 ∈ 𝑥 ( O ‘𝑦) ∈ Fin)
30	29	3adant2 1137	. . . . 5 ⊢ ((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (( O ‘𝑦) ∈ Fin → 𝑦 ∈ ω) ∧ ( O ‘𝑥) ∈ Fin) → ∀𝑦 ∈ 𝑥 ( O ‘𝑦) ∈ Fin)
31		r19.26 3099	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝑥 ((( O ‘𝑦) ∈ Fin → 𝑦 ∈ ω) ∧ ( O ‘𝑦) ∈ Fin) ↔ (∀𝑦 ∈ 𝑥 (( O ‘𝑦) ∈ Fin → 𝑦 ∈ ω) ∧ ∀𝑦 ∈ 𝑥 ( O ‘𝑦) ∈ Fin))
32		pm2.27 42	. . . . . . . . . . . . 13 ⊢ (( O ‘𝑦) ∈ Fin → ((( O ‘𝑦) ∈ Fin → 𝑦 ∈ ω) → 𝑦 ∈ ω))
33	32	impcom 408	. . . . . . . . . . . 12 ⊢ (((( O ‘𝑦) ∈ Fin → 𝑦 ∈ ω) ∧ ( O ‘𝑦) ∈ Fin) → 𝑦 ∈ ω)
34	33	ralimi 3076	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ 𝑥 ((( O ‘𝑦) ∈ Fin → 𝑦 ∈ ω) ∧ ( O ‘𝑦) ∈ Fin) → ∀𝑦 ∈ 𝑥 𝑦 ∈ ω)
35		dfss3 3904	. . . . . . . . . . 11 ⊢ (𝑥 ⊆ ω ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ ω)
36	34, 35	sylibr 235	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝑥 ((( O ‘𝑦) ∈ Fin → 𝑦 ∈ ω) ∧ ( O ‘𝑦) ∈ Fin) → 𝑥 ⊆ ω)
37	31, 36	sylbir 236	. . . . . . . . 9 ⊢ ((∀𝑦 ∈ 𝑥 (( O ‘𝑦) ∈ Fin → 𝑦 ∈ ω) ∧ ∀𝑦 ∈ 𝑥 ( O ‘𝑦) ∈ Fin) → 𝑥 ⊆ ω)
38		eloni 6320	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → Ord 𝑥)
39		ordom 7816	. . . . . . . . . . . . 13 ⊢ Ord ω
40		ordsseleq 6339	. . . . . . . . . . . . 13 ⊢ ((Ord 𝑥 ∧ Ord ω) → (𝑥 ⊆ ω ↔ (𝑥 ∈ ω ∨ 𝑥 = ω)))
41	39, 40	mpan2 697	. . . . . . . . . . . 12 ⊢ (Ord 𝑥 → (𝑥 ⊆ ω ↔ (𝑥 ∈ ω ∨ 𝑥 = ω)))
42	38, 41	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → (𝑥 ⊆ ω ↔ (𝑥 ∈ ω ∨ 𝑥 = ω)))
43	42	adantr 481	. . . . . . . . . 10 ⊢ ((𝑥 ∈ On ∧ ( O ‘𝑥) ∈ Fin) → (𝑥 ⊆ ω ↔ (𝑥 ∈ ω ∨ 𝑥 = ω)))
44		fveq2 6827	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ω → ( O ‘𝑥) = ( O ‘ω))
45		eqvisset 3451	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = ω → ω ∈ V)
46		bdayfun 27758	. . . . . . . . . . . . . . . . . . . 20 ⊢ Fun bday
47		n0sexg 28326	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ω ∈ V → ℕ0s ∈ V)
48		resfunexg 7159	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((Fun bday ∧ ℕ0s ∈ V) → ( bday ↾ ℕ0s) ∈ V)
49	46, 47, 48	sylancr 593	. . . . . . . . . . . . . . . . . . 19 ⊢ (ω ∈ V → ( bday ↾ ℕ0s) ∈ V)
50		cnvexg 7864	. . . . . . . . . . . . . . . . . . 19 ⊢ (( bday ↾ ℕ0s) ∈ V → ◡( bday ↾ ℕ0s) ∈ V)
51	49, 50	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (ω ∈ V → ◡( bday ↾ ℕ0s) ∈ V)
52		bdayn0sf1o 28380	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ( bday ↾ ℕ0s):ℕ0s–1-1-onto→ω
53	52	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ω ∈ V → ( bday ↾ ℕ0s):ℕ0s–1-1-onto→ω)
54		f1ocnv 6779	. . . . . . . . . . . . . . . . . . . 20 ⊢ (( bday ↾ ℕ0s):ℕ0s–1-1-onto→ω → ◡( bday ↾ ℕ0s):ω–1-1-onto→ℕ0s)
55		f1of1 6766	. . . . . . . . . . . . . . . . . . . 20 ⊢ (◡( bday ↾ ℕ0s):ω–1-1-onto→ℕ0s → ◡( bday ↾ ℕ0s):ω–1-1→ℕ0s)
56	53, 54, 55	3syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (ω ∈ V → ◡( bday ↾ ℕ0s):ω–1-1→ℕ0s)
57		n0ssoldg 28363	. . . . . . . . . . . . . . . . . . 19 ⊢ (ω ∈ V → ℕ0s ⊆ ( O ‘ω))
58		f1ss 6728	. . . . . . . . . . . . . . . . . . 19 ⊢ ((◡( bday ↾ ℕ0s):ω–1-1→ℕ0s ∧ ℕ0s ⊆ ( O ‘ω)) → ◡( bday ↾ ℕ0s):ω–1-1→( O ‘ω))
59	56, 57, 58	syl2anc 590	. . . . . . . . . . . . . . . . . 18 ⊢ (ω ∈ V → ◡( bday ↾ ℕ0s):ω–1-1→( O ‘ω))
60		f1eq1 6718	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = ◡( bday ↾ ℕ0s) → (𝑓:ω–1-1→( O ‘ω) ↔ ◡( bday ↾ ℕ0s):ω–1-1→( O ‘ω)))
61	51, 59, 60	spcedv 3536	. . . . . . . . . . . . . . . . 17 ⊢ (ω ∈ V → ∃𝑓 𝑓:ω–1-1→( O ‘ω))
62		fvex 6840	. . . . . . . . . . . . . . . . . 18 ⊢ ( O ‘ω) ∈ V
63	62	brdom 8897	. . . . . . . . . . . . . . . . 17 ⊢ (ω ≼ ( O ‘ω) ↔ ∃𝑓 𝑓:ω–1-1→( O ‘ω))
64	61, 63	sylibr 235	. . . . . . . . . . . . . . . 16 ⊢ (ω ∈ V → ω ≼ ( O ‘ω))
65		infinfg 10479	. . . . . . . . . . . . . . . . 17 ⊢ ((ω ∈ V ∧ ( O ‘ω) ∈ V) → (¬ ( O ‘ω) ∈ Fin ↔ ω ≼ ( O ‘ω)))
66	62, 65	mpan2 697	. . . . . . . . . . . . . . . 16 ⊢ (ω ∈ V → (¬ ( O ‘ω) ∈ Fin ↔ ω ≼ ( O ‘ω)))
67	64, 66	mpbird 258	. . . . . . . . . . . . . . 15 ⊢ (ω ∈ V → ¬ ( O ‘ω) ∈ Fin)
68	45, 67	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ω → ¬ ( O ‘ω) ∈ Fin)
69	44, 68	eqneltrd 2859	. . . . . . . . . . . . 13 ⊢ (𝑥 = ω → ¬ ( O ‘𝑥) ∈ Fin)
70	69	con2i 139	. . . . . . . . . . . 12 ⊢ (( O ‘𝑥) ∈ Fin → ¬ 𝑥 = ω)
71	70	adantl 482	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ On ∧ ( O ‘𝑥) ∈ Fin) → ¬ 𝑥 = ω)
72		orel2 896	. . . . . . . . . . 11 ⊢ (¬ 𝑥 = ω → ((𝑥 ∈ ω ∨ 𝑥 = ω) → 𝑥 ∈ ω))
73	71, 72	syl 17	. . . . . . . . . 10 ⊢ ((𝑥 ∈ On ∧ ( O ‘𝑥) ∈ Fin) → ((𝑥 ∈ ω ∨ 𝑥 = ω) → 𝑥 ∈ ω))
74	43, 73	sylbid 241	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ ( O ‘𝑥) ∈ Fin) → (𝑥 ⊆ ω → 𝑥 ∈ ω))
75	37, 74	syl5 34	. . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ ( O ‘𝑥) ∈ Fin) → ((∀𝑦 ∈ 𝑥 (( O ‘𝑦) ∈ Fin → 𝑦 ∈ ω) ∧ ∀𝑦 ∈ 𝑥 ( O ‘𝑦) ∈ Fin) → 𝑥 ∈ ω))
76	75	expd 416	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ ( O ‘𝑥) ∈ Fin) → (∀𝑦 ∈ 𝑥 (( O ‘𝑦) ∈ Fin → 𝑦 ∈ ω) → (∀𝑦 ∈ 𝑥 ( O ‘𝑦) ∈ Fin → 𝑥 ∈ ω)))
77	76	3impia 1123	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ ( O ‘𝑥) ∈ Fin ∧ ∀𝑦 ∈ 𝑥 (( O ‘𝑦) ∈ Fin → 𝑦 ∈ ω)) → (∀𝑦 ∈ 𝑥 ( O ‘𝑦) ∈ Fin → 𝑥 ∈ ω))
78	77	3com23 1132	. . . . 5 ⊢ ((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (( O ‘𝑦) ∈ Fin → 𝑦 ∈ ω) ∧ ( O ‘𝑥) ∈ Fin) → (∀𝑦 ∈ 𝑥 ( O ‘𝑦) ∈ Fin → 𝑥 ∈ ω))
79	30, 78	mpd 15	. . . 4 ⊢ ((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (( O ‘𝑦) ∈ Fin → 𝑦 ∈ ω) ∧ ( O ‘𝑥) ∈ Fin) → 𝑥 ∈ ω)
80	79	3exp 1125	. . 3 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 (( O ‘𝑦) ∈ Fin → 𝑦 ∈ ω) → (( O ‘𝑥) ∈ Fin → 𝑥 ∈ ω)))
81	5, 9, 80	tfis3 7798	. 2 ⊢ (𝐴 ∈ On → (( O ‘𝐴) ∈ Fin → 𝐴 ∈ ω))
82	1, 81	impbid2 227	1 ⊢ (𝐴 ∈ On → (𝐴 ∈ ω ↔ ( O ‘𝐴) ∈ Fin))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 853   ∧ w3a 1092   = wceq 1547  ∃wex 1786   ∈ wcel 2119  ∀wral 3053  Vcvv 3431   ⊆ wss 3883  𝒫 cpw 4529  ∪ cuni 4838   class class class wbr 5072  ^◡ccnv 5617  dom cdm 5618   ↾ cres 5620   “ cima 5621  Ord word 6309  Oncon0 6310  Fun wfun 6479  ⟶wf 6481  –1-1→wf1 6482  –1-1-onto→wf1o 6484  ‘cfv 6485  ωcom 7806   ≼ cdom 8881  Fincfn 8883   No csur 27621   bday cbday 27623   M cmade 27832   O cold 27833  ℕ_0scn0s 28322

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-ac2 10376

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-tp 4560  df-op 4562  df-ot 4564  df-uni 4839  df-int 4878  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-se 5572  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-isom 6494  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-nadd 8592  df-er 8633  df-map 8765  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9854  df-acn 9857  df-ac 10029  df-no 27624  df-lts 27625  df-bday 27626  df-les 27727  df-slts 27768  df-cuts 27770  df-0s 27817  df-1s 27818  df-made 27837  df-old 27838  df-left 27840  df-right 27841  df-norec 27948  df-norec2 27959  df-adds 27970  df-negs 28031  df-subs 28032  df-ons 28262  df-n0s 28324

This theorem is referenced by: (None)