Theorem oldfib 28820

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 28357	. 2 ⊢ (𝐴 ∈ ω → (O‘𝐴) ∈ Fin)
2		fveq2 6916	. . . . 5 ⊢ (𝑥 = 𝑦 → (O‘𝑥) = (O‘𝑦))
3	2	eleq1d 2849	. . . 4 ⊢ (𝑥 = 𝑦 → ((O‘𝑥) ∈ Fin ↔ (O‘𝑦) ∈ Fin))
4		eleq1 2852	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ ω ↔ 𝑦 ∈ ω))
5	3, 4	imbi12d 347	. . 3 ⊢ (𝑥 = 𝑦 → (((O‘𝑥) ∈ Fin → 𝑥 ∈ ω) ↔ ((O‘𝑦) ∈ Fin → 𝑦 ∈ ω)))
6		fveq2 6916	. . . . 5 ⊢ (𝑥 = 𝐴 → (O‘𝑥) = (O‘𝐴))
7	6	eleq1d 2849	. . . 4 ⊢ (𝑥 = 𝐴 → ((O‘𝑥) ∈ Fin ↔ (O‘𝐴) ∈ Fin))
8		eleq1 2852	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ ω ↔ 𝐴 ∈ ω))
9	7, 8	imbi12d 347	. . 3 ⊢ (𝑥 = 𝐴 → (((O‘𝑥) ∈ Fin → 𝑥 ∈ ω) ↔ ((O‘𝐴) ∈ Fin → 𝐴 ∈ ω)))
10		oldval 28277	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → (O‘𝑥) = ∪ (M “ 𝑥))
11	10	eleq1d 2849	. . . . . . . . . 10 ⊢ (𝑥 ∈ On → ((O‘𝑥) ∈ Fin ↔ ∪ (M “ 𝑥) ∈ Fin))
12	11	biimpa 482	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ (O‘𝑥) ∈ Fin) → ∪ (M “ 𝑥) ∈ Fin)
13		unifi3 9389	. . . . . . . . 9 ⊢ (∪ (M “ 𝑥) ∈ Fin → (M “ 𝑥) ⊆ Fin)
14	12, 13	syl 18	. . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ (O‘𝑥) ∈ Fin) → (M “ 𝑥) ⊆ Fin)
15		madef 28279	. . . . . . . . . . 11 ⊢ M:On⟶𝒫 No
16		ffun 6742	. . . . . . . . . . 11 ⊢ (M:On⟶𝒫 No → Fun M)
17	15, 16	ax-mp 5	. . . . . . . . . 10 ⊢ Fun M
18		onss 7833	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → 𝑥 ⊆ On)
19	15	fdmi 6751	. . . . . . . . . . 11 ⊢ dom M = On
20	18, 19	sseqtrrdi 3954	. . . . . . . . . 10 ⊢ (𝑥 ∈ On → 𝑥 ⊆ dom M)
21		funimass4 6980	. . . . . . . . . 10 ⊢ ((Fun M ∧ 𝑥 ⊆ dom M) → ((M “ 𝑥) ⊆ Fin ↔ ∀𝑦 ∈ 𝑥 (M‘𝑦) ∈ Fin))
22	17, 20, 21	sylancr 599	. . . . . . . . 9 ⊢ (𝑥 ∈ On → ((M “ 𝑥) ⊆ Fin ↔ ∀𝑦 ∈ 𝑥 (M‘𝑦) ∈ Fin))
23	22	adantr 486	. . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ (O‘𝑥) ∈ Fin) → ((M “ 𝑥) ⊆ Fin ↔ ∀𝑦 ∈ 𝑥 (M‘𝑦) ∈ Fin))
24	14, 23	mpbid 235	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ (O‘𝑥) ∈ Fin) → ∀𝑦 ∈ 𝑥 (M‘𝑦) ∈ Fin)
25		oldssmade 28310	. . . . . . . . 9 ⊢ (O‘𝑦) ⊆ (M‘𝑦)
26		ssfi 9226	. . . . . . . . 9 ⊢ (((M‘𝑦) ∈ Fin ∧ (O‘𝑦) ⊆ (M‘𝑦)) → (O‘𝑦) ∈ Fin)
27	25, 26	mpan2 704	. . . . . . . 8 ⊢ ((M‘𝑦) ∈ Fin → (O‘𝑦) ∈ Fin)
28	27	ralimi 3103	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝑥 (M‘𝑦) ∈ Fin → ∀𝑦 ∈ 𝑥 (O‘𝑦) ∈ Fin)
29	24, 28	syl 18	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ (O‘𝑥) ∈ Fin) → ∀𝑦 ∈ 𝑥 (O‘𝑦) ∈ Fin)
30	29	3adant2 1149	. . . . 5 ⊢ ((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 ((O‘𝑦) ∈ Fin → 𝑦 ∈ ω) ∧ (O‘𝑥) ∈ Fin) → ∀𝑦 ∈ 𝑥 (O‘𝑦) ∈ Fin)
31		r19.26 3126	. . . . . . . . . 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 43	. . . . . . . . . . . . 13 ⊢ ((O‘𝑦) ∈ Fin → (((O‘𝑦) ∈ Fin → 𝑦 ∈ ω) → 𝑦 ∈ ω))
33	32	impcom 413	. . . . . . . . . . . 12 ⊢ ((((O‘𝑦) ∈ Fin → 𝑦 ∈ ω) ∧ (O‘𝑦) ∈ Fin) → 𝑦 ∈ ω)
34	33	ralimi 3103	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ 𝑥 (((O‘𝑦) ∈ Fin → 𝑦 ∈ ω) ∧ (O‘𝑦) ∈ Fin) → ∀𝑦 ∈ 𝑥 𝑦 ∈ ω)
35		dfss3 3902	. . . . . . . . . . 11 ⊢ (𝑥 ⊆ ω ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ ω)
36	34, 35	sylibr 237	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝑥 (((O‘𝑦) ∈ Fin → 𝑦 ∈ ω) ∧ (O‘𝑦) ∈ Fin) → 𝑥 ⊆ ω)
37	31, 36	sylbir 238	. . . . . . . . 9 ⊢ ((∀𝑦 ∈ 𝑥 ((O‘𝑦) ∈ Fin → 𝑦 ∈ ω) ∧ ∀𝑦 ∈ 𝑥 (O‘𝑦) ∈ Fin) → 𝑥 ⊆ ω)
38		eloni 6398	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → Ord 𝑥)
39		ordom 7921	. . . . . . . . . . . . 13 ⊢ Ord ω
40		ordsseleq 6418	. . . . . . . . . . . . 13 ⊢ ((Ord 𝑥 ∧ Ord ω) → (𝑥 ⊆ ω ↔ (𝑥 ∈ ω ∨ 𝑥 = ω)))
41	39, 40	mpan2 704	. . . . . . . . . . . 12 ⊢ (Ord 𝑥 → (𝑥 ⊆ ω ↔ (𝑥 ∈ ω ∨ 𝑥 = ω)))
42	38, 41	syl 18	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → (𝑥 ⊆ ω ↔ (𝑥 ∈ ω ∨ 𝑥 = ω)))
43	42	adantr 486	. . . . . . . . . 10 ⊢ ((𝑥 ∈ On ∧ (O‘𝑥) ∈ Fin) → (𝑥 ⊆ ω ↔ (𝑥 ∈ ω ∨ 𝑥 = ω)))
44		fveq2 6916	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ω → (O‘𝑥) = (O‘ω))
45		eqvisset 3474	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = ω → ω ∈ V)
46		bdayfun 28190	. . . . . . . . . . . . . . . . . . . 20 ⊢ Fun bday
47		n0sexg 28759	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ω ∈ V → ℕ0s ∈ V)
48		resfunexg 7252	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((Fun bday ∧ ℕ0s ∈ V) → (bday ↾ ℕ0s) ∈ V)
49	46, 47, 48	sylancr 599	. . . . . . . . . . . . . . . . . . 19 ⊢ (ω ∈ V → (bday ↾ ℕ0s) ∈ V)
50		cnvexg 7970	. . . . . . . . . . . . . . . . . . 19 ⊢ ((bday ↾ ℕ0s) ∈ V → ◡(bday ↾ ℕ0s) ∈ V)
51	49, 50	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (ω ∈ V → ◡(bday ↾ ℕ0s) ∈ V)
52		bdayn0sf1o 28813	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (bday ↾ ℕ0s):ℕ0s⟶1-1ontoω
53	52	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ω ∈ V → (bday ↾ ℕ0s):ℕ0s⟶1-1ontoω)
54		f1ocnv 6867	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((bday ↾ ℕ0s):ℕ0s⟶1-1ontoω → ◡(bday ↾ ℕ0s):ω⟶1-1ontoℕ0s)
55		f1of1 6853	. . . . . . . . . . . . . . . . . . . 20 ⊢ (◡(bday ↾ ℕ0s):ω⟶1-1ontoℕ0s → ◡(bday ↾ ℕ0s):ω⟶1-1ℕ0s)
56	53, 54, 55	3syl 19	. . . . . . . . . . . . . . . . . . 19 ⊢ (ω ∈ V → ◡(bday ↾ ℕ0s):ω⟶1-1ℕ0s)
57		n0ssoldg 28796	. . . . . . . . . . . . . . . . . . 19 ⊢ (ω ∈ V → ℕ0s ⊆ (O‘ω))
58		f1ss 6815	. . . . . . . . . . . . . . . . . . 19 ⊢ ((◡(bday ↾ ℕ0s):ω⟶1-1ℕ0s ∧ ℕ0s ⊆ (O‘ω)) → ◡(bday ↾ ℕ0s):ω⟶1-1(O‘ω))
59	56, 57, 58	syl2anc 596	. . . . . . . . . . . . . . . . . 18 ⊢ (ω ∈ V → ◡(bday ↾ ℕ0s):ω⟶1-1(O‘ω))
60		f1eq1 6803	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = ◡(bday ↾ ℕ0s) → (𝑓:ω⟶1-1(O‘ω) ↔ ◡(bday ↾ ℕ0s):ω⟶1-1(O‘ω)))
61	51, 59, 60	spcedv 3556	. . . . . . . . . . . . . . . . 17 ⊢ (ω ∈ V → ∃𝑓 𝑓:ω⟶1-1(O‘ω))
62		fvex 6929	. . . . . . . . . . . . . . . . . 18 ⊢ (O‘ω) ∈ V
63	62	brdom 9025	. . . . . . . . . . . . . . . . 17 ⊢ (ω ≼ (O‘ω) ↔ ∃𝑓 𝑓:ω⟶1-1(O‘ω))
64	61, 63	sylibr 237	. . . . . . . . . . . . . . . 16 ⊢ (ω ∈ V → ω ≼ (O‘ω))
65		infinfg 10689	. . . . . . . . . . . . . . . . 17 ⊢ ((ω ∈ V ∧ (O‘ω) ∈ V) → (¬ (O‘ω) ∈ Fin ↔ ω ≼ (O‘ω)))
66	62, 65	mpan2 704	. . . . . . . . . . . . . . . 16 ⊢ (ω ∈ V → (¬ (O‘ω) ∈ Fin ↔ ω ≼ (O‘ω)))
67	64, 66	mpbird 260	. . . . . . . . . . . . . . 15 ⊢ (ω ∈ V → ¬ (O‘ω) ∈ Fin)
68	45, 67	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ω → ¬ (O‘ω) ∈ Fin)
69	44, 68	eqneltrd 2884	. . . . . . . . . . . . 13 ⊢ (𝑥 = ω → ¬ (O‘𝑥) ∈ Fin)
70	69	con2i 140	. . . . . . . . . . . 12 ⊢ ((O‘𝑥) ∈ Fin → ¬ 𝑥 = ω)
71	70	adantl 487	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ On ∧ (O‘𝑥) ∈ Fin) → ¬ 𝑥 = ω)
72		orel2 904	. . . . . . . . . . 11 ⊢ (¬ 𝑥 = ω → ((𝑥 ∈ ω ∨ 𝑥 = ω) → 𝑥 ∈ ω))
73	71, 72	syl 18	. . . . . . . . . 10 ⊢ ((𝑥 ∈ On ∧ (O‘𝑥) ∈ Fin) → ((𝑥 ∈ ω ∨ 𝑥 = ω) → 𝑥 ∈ ω))
74	43, 73	sylbid 243	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ (O‘𝑥) ∈ Fin) → (𝑥 ⊆ ω → 𝑥 ∈ ω))
75	37, 74	syl5 35	. . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ (O‘𝑥) ∈ Fin) → ((∀𝑦 ∈ 𝑥 ((O‘𝑦) ∈ Fin → 𝑦 ∈ ω) ∧ ∀𝑦 ∈ 𝑥 (O‘𝑦) ∈ Fin) → 𝑥 ∈ ω))
76	75	expd 421	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ (O‘𝑥) ∈ Fin) → (∀𝑦 ∈ 𝑥 ((O‘𝑦) ∈ Fin → 𝑦 ∈ ω) → (∀𝑦 ∈ 𝑥 (O‘𝑦) ∈ Fin → 𝑥 ∈ ω)))
77	76	3impia 1135	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ (O‘𝑥) ∈ Fin ∧ ∀𝑦 ∈ 𝑥 ((O‘𝑦) ∈ Fin → 𝑦 ∈ ω)) → (∀𝑦 ∈ 𝑥 (O‘𝑦) ∈ Fin → 𝑥 ∈ ω))
78	77	3com23 1144	. . . . 5 ⊢ ((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 ((O‘𝑦) ∈ Fin → 𝑦 ∈ ω) ∧ (O‘𝑥) ∈ Fin) → (∀𝑦 ∈ 𝑥 (O‘𝑦) ∈ Fin → 𝑥 ∈ ω))
79	30, 78	mpd 16	. . . 4 ⊢ ((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 ((O‘𝑦) ∈ Fin → 𝑦 ∈ ω) ∧ (O‘𝑥) ∈ Fin) → 𝑥 ∈ ω)
80	79	3exp 1137	. . 3 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 ((O‘𝑦) ∈ Fin → 𝑦 ∈ ω) → ((O‘𝑥) ∈ Fin → 𝑥 ∈ ω)))
81	5, 9, 80	tfis3 7903	. 2 ⊢ (𝐴 ∈ On → ((O‘𝐴) ∈ Fin → 𝐴 ∈ ω))
82	1, 81	impbid2 229	1 ⊢ (𝐴 ∈ On → (𝐴 ∈ ω ↔ (O‘𝐴) ∈ Fin))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2147  ∀wral 3080  Vcvv 3454   ⊆ wss 3900  𝒫 cpw 4560  ∪ cuni 4872   class class class wbr 5108  ^◡ccnv 5657  dom cdm 5658   ↾ cres 5660   “ cima 5661  Ord word 6387  Oncon0 6388  Fun wfun 6560  ⟶wf 6562  ⟶1-1wf1 6563  ⟶1-1ontowf1o 6565  ‘cfv 6566  ωcom 7911   ≼ cdom 9009  Fincfn 9011  Nocsur 28054  bdaycbday 28056  Mcmade 28265  Ocold 28266  ℕ_0scn0s 28755

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-ac2 10580

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4841  df-ot 4864  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-isom 6575  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-2o 8515  df-nadd 8713  df-er 8755  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-card 10057  df-ac 10234  df-no 28057  df-lts 28058  df-bday 28059  df-les 28159  df-slts 28201  df-cuts 28203  df-0s 28250  df-1s 28251  df-made 28270  df-old 28271  df-left 28273  df-right 28274  df-norec 28381  df-norec2 28392  df-adds 28403  df-negs 28464  df-subs 28465  df-ons 28695  df-n0s 28757

This theorem is used by: (None)