Theorem oldfib 28756

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 28293	. 2 ⊢ (𝐴 ∈ ω → ( O ‘𝐴) ∈ Fin)
2		fveq2 6883	. . . . 5 ⊢ (𝑥 = 𝑦 → ( O ‘𝑥) = ( O ‘𝑦))
3	2	eleq1d 2846	. . . 4 ⊢ (𝑥 = 𝑦 → (( O ‘𝑥) ∈ Fin ↔ ( O ‘𝑦) ∈ Fin))
4		eleq1 2849	. . . 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 6883	. . . . 5 ⊢ (𝑥 = 𝐴 → ( O ‘𝑥) = ( O ‘𝐴))
7	6	eleq1d 2846	. . . 4 ⊢ (𝑥 = 𝐴 → (( O ‘𝑥) ∈ Fin ↔ ( O ‘𝐴) ∈ Fin))
8		eleq1 2849	. . . 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 28213	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → ( O ‘𝑥) = ∪ ( M “ 𝑥))
11	10	eleq1d 2846	. . . . . . . . . 10 ⊢ (𝑥 ∈ On → (( O ‘𝑥) ∈ Fin ↔ ∪ ( M “ 𝑥) ∈ Fin))
12	11	biimpa 482	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ ( O ‘𝑥) ∈ Fin) → ∪ ( M “ 𝑥) ∈ Fin)
13		unifi3 9344	. . . . . . . . 9 ⊢ (∪ ( M “ 𝑥) ∈ Fin → ( M “ 𝑥) ⊆ Fin)
14	12, 13	syl 18	. . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ ( O ‘𝑥) ∈ Fin) → ( M “ 𝑥) ⊆ Fin)
15		madef 28215	. . . . . . . . . . 11 ⊢ M :On⟶𝒫 No
16		ffun 6710	. . . . . . . . . . 11 ⊢ ( M :On⟶𝒫 No → Fun M )
17	15, 16	ax-mp 5	. . . . . . . . . 10 ⊢ Fun M
18		onss 7797	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → 𝑥 ⊆ On)
19	15	fdmi 6719	. . . . . . . . . . 11 ⊢ dom M = On
20	18, 19	sseqtrrdi 3972	. . . . . . . . . 10 ⊢ (𝑥 ∈ On → 𝑥 ⊆ dom M )
21		funimass4 6947	. . . . . . . . . 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 28246	. . . . . . . . 9 ⊢ ( O ‘𝑦) ⊆ ( M ‘𝑦)
26		ssfi 9181	. . . . . . . . 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 3100	. . . . . . 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 3123	. . . . . . . . . 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 3100	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ 𝑥 ((( O ‘𝑦) ∈ Fin → 𝑦 ∈ ω) ∧ ( O ‘𝑦) ∈ Fin) → ∀𝑦 ∈ 𝑥 𝑦 ∈ ω)
35		dfss3 3920	. . . . . . . . . . 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 6371	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → Ord 𝑥)
39		ordom 7885	. . . . . . . . . . . . 13 ⊢ Ord ω
40		ordsseleq 6391	. . . . . . . . . . . . 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 6883	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ω → ( O ‘𝑥) = ( O ‘ω))
45		eqvisset 3471	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = ω → ω ∈ V)
46		bdayfun 28126	. . . . . . . . . . . . . . . . . . . 20 ⊢ Fun bday
47		n0sexg 28695	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ω ∈ V → ℕ0s ∈ V)
48		resfunexg 7219	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((Fun bday ∧ ℕ0s ∈ V) → ( bday ↾ ℕ0s) ∈ V)
49	46, 47, 48	sylancr 599	. . . . . . . . . . . . . . . . . . 19 ⊢ (ω ∈ V → ( bday ↾ ℕ0s) ∈ V)
50		cnvexg 7934	. . . . . . . . . . . . . . . . . . 19 ⊢ (( bday ↾ ℕ0s) ∈ V → ◡( bday ↾ ℕ0s) ∈ V)
51	49, 50	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (ω ∈ V → ◡( bday ↾ ℕ0s) ∈ V)
52		bdayn0sf1o 28749	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ( bday ↾ ℕ0s):ℕ0s–1-1-onto→ω
53	52	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ω ∈ V → ( bday ↾ ℕ0s):ℕ0s–1-1-onto→ω)
54		f1ocnv 6835	. . . . . . . . . . . . . . . . . . . 20 ⊢ (( bday ↾ ℕ0s):ℕ0s–1-1-onto→ω → ◡( bday ↾ ℕ0s):ω–1-1-onto→ℕ0s)
55		f1of1 6821	. . . . . . . . . . . . . . . . . . . 20 ⊢ (◡( bday ↾ ℕ0s):ω–1-1-onto→ℕ0s → ◡( bday ↾ ℕ0s):ω–1-1→ℕ0s)
56	53, 54, 55	3syl 19	. . . . . . . . . . . . . . . . . . 19 ⊢ (ω ∈ V → ◡( bday ↾ ℕ0s):ω–1-1→ℕ0s)
57		n0ssoldg 28732	. . . . . . . . . . . . . . . . . . 19 ⊢ (ω ∈ V → ℕ0s ⊆ ( O ‘ω))
58		f1ss 6783	. . . . . . . . . . . . . . . . . . 19 ⊢ ((◡( bday ↾ ℕ0s):ω–1-1→ℕ0s ∧ 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 6771	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = ◡( bday ↾ ℕ0s) → (𝑓:ω–1-1→( O ‘ω) ↔ ◡( bday ↾ ℕ0s):ω–1-1→( O ‘ω)))
61	51, 59, 60	spcedv 3553	. . . . . . . . . . . . . . . . 17 ⊢ (ω ∈ V → ∃𝑓 𝑓:ω–1-1→( O ‘ω))
62		fvex 6896	. . . . . . . . . . . . . . . . . 18 ⊢ ( O ‘ω) ∈ V
63	62	brdom 8980	. . . . . . . . . . . . . . . . 17 ⊢ (ω ≼ ( O ‘ω) ↔ ∃𝑓 𝑓:ω–1-1→( O ‘ω))
64	61, 63	sylibr 237	. . . . . . . . . . . . . . . 16 ⊢ (ω ∈ V → ω ≼ ( O ‘ω))
65		infinfg 10643	. . . . . . . . . . . . . . . . 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 2881	. . . . . . . . . . . . 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 7867	. 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 2145  ∀wral 3077  Vcvv 3451   ⊆ wss 3899  𝒫 cpw 4557  ∪ cuni 4867   class class class wbr 5103  ^◡ccnv 5650  dom cdm 5651   ↾ cres 5653   “ cima 5654  Ord word 6360  Oncon0 6361  Fun wfun 6531  ⟶wf 6533  –1-1→wf1 6534  –1-1-onto→wf1o 6536  ‘cfv 6537  ωcom 7875   ≼ cdom 8964  Fincfn 8966   No csur 27990   bday cbday 27992   M cmade 28201   O cold 28202  ℕ_0scn0s 28691

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-ac2 10534

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 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-ot 4593  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-om 7876  df-1st 7999  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-1o 8469  df-2o 8470  df-nadd 8668  df-er 8710  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-card 10013  df-ac 10188  df-no 27993  df-lts 27994  df-bday 27995  df-les 28095  df-slts 28137  df-cuts 28139  df-0s 28186  df-1s 28187  df-made 28206  df-old 28207  df-left 28209  df-right 28210  df-norec 28317  df-norec2 28328  df-adds 28339  df-negs 28400  df-subs 28401  df-ons 28631  df-n0s 28693

This theorem is used by: (None)