Theorem oldfib 28772

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 28309	. 2 ⊢ (𝐴 ∈ ω → (O‘𝐴) ∈ Fin)
2		fveq2 6890	. . . . 5 ⊢ (𝑥 = 𝑦 → (O‘𝑥) = (O‘𝑦))
3	2	eleq1d 2845	. . . 4 ⊢ (𝑥 = 𝑦 → ((O‘𝑥) ∈ Fin ↔ (O‘𝑦) ∈ Fin))
4		eleq1 2848	. . . 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 6890	. . . . 5 ⊢ (𝑥 = 𝐴 → (O‘𝑥) = (O‘𝐴))
7	6	eleq1d 2845	. . . 4 ⊢ (𝑥 = 𝐴 → ((O‘𝑥) ∈ Fin ↔ (O‘𝐴) ∈ Fin))
8		eleq1 2848	. . . 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 28229	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → (O‘𝑥) = ∪ (M “ 𝑥))
11	10	eleq1d 2845	. . . . . . . . . 10 ⊢ (𝑥 ∈ On → ((O‘𝑥) ∈ Fin ↔ ∪ (M “ 𝑥) ∈ Fin))
12	11	biimpa 482	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ (O‘𝑥) ∈ Fin) → ∪ (M “ 𝑥) ∈ Fin)
13		unifi3 9358	. . . . . . . . 9 ⊢ (∪ (M “ 𝑥) ∈ Fin → (M “ 𝑥) ⊆ Fin)
14	12, 13	syl 18	. . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ (O‘𝑥) ∈ Fin) → (M “ 𝑥) ⊆ Fin)
15		madef 28231	. . . . . . . . . . 11 ⊢ M:On⟶𝒫 No
16		ffun 6717	. . . . . . . . . . 11 ⊢ (M:On⟶𝒫 No → Fun M)
17	15, 16	ax-mp 5	. . . . . . . . . 10 ⊢ Fun M
18		onss 7803	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → 𝑥 ⊆ On)
19	15	fdmi 6726	. . . . . . . . . . 11 ⊢ dom M = On
20	18, 19	sseqtrrdi 3950	. . . . . . . . . 10 ⊢ (𝑥 ∈ On → 𝑥 ⊆ dom M)
21		funimass4 6954	. . . . . . . . . 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 28262	. . . . . . . . 9 ⊢ (O‘𝑦) ⊆ (M‘𝑦)
26		ssfi 9195	. . . . . . . . 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 3099	. . . . . . 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 3122	. . . . . . . . . 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 3099	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ 𝑥 (((O‘𝑦) ∈ Fin → 𝑦 ∈ ω) ∧ (O‘𝑦) ∈ Fin) → ∀𝑦 ∈ 𝑥 𝑦 ∈ ω)
35		dfss3 3898	. . . . . . . . . . 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 6375	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → Ord 𝑥)
39		ordom 7891	. . . . . . . . . . . . 13 ⊢ Ord ω
40		ordsseleq 6395	. . . . . . . . . . . . 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 6890	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ω → (O‘𝑥) = (O‘ω))
45		eqvisset 3470	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = ω → ω ∈ V)
46		bdayfun 28142	. . . . . . . . . . . . . . . . . . . 20 ⊢ Fun bday
47		n0sexg 28711	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ω ∈ V → ℕ0s ∈ V)
48		resfunexg 7225	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((Fun bday ∧ ℕ0s ∈ V) → (bday ↾ ℕ0s) ∈ V)
49	46, 47, 48	sylancr 599	. . . . . . . . . . . . . . . . . . 19 ⊢ (ω ∈ V → (bday ↾ ℕ0s) ∈ V)
50		cnvexg 7940	. . . . . . . . . . . . . . . . . . 19 ⊢ ((bday ↾ ℕ0s) ∈ V → ◡(bday ↾ ℕ0s) ∈ V)
51	49, 50	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (ω ∈ V → ◡(bday ↾ ℕ0s) ∈ V)
52		bdayn0sf1o 28765	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (bday ↾ ℕ0s):ℕ0s–1-1-onto→ω
53	52	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ω ∈ V → (bday ↾ ℕ0s):ℕ0s–1-1-onto→ω)
54		f1ocnv 6842	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((bday ↾ ℕ0s):ℕ0s–1-1-onto→ω → ◡(bday ↾ ℕ0s):ω–1-1-onto→ℕ0s)
55		f1of1 6828	. . . . . . . . . . . . . . . . . . . 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 28748	. . . . . . . . . . . . . . . . . . 19 ⊢ (ω ∈ V → ℕ0s ⊆ (O‘ω))
58		f1ss 6790	. . . . . . . . . . . . . . . . . . 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 6778	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = ◡(bday ↾ ℕ0s) → (𝑓:ω–1-1→(O‘ω) ↔ ◡(bday ↾ ℕ0s):ω–1-1→(O‘ω)))
61	51, 59, 60	spcedv 3552	. . . . . . . . . . . . . . . . 17 ⊢ (ω ∈ V → ∃𝑓 𝑓:ω–1-1→(O‘ω))
62		fvex 6903	. . . . . . . . . . . . . . . . . 18 ⊢ (O‘ω) ∈ V
63	62	brdom 8994	. . . . . . . . . . . . . . . . 17 ⊢ (ω ≼ (O‘ω) ↔ ∃𝑓 𝑓:ω–1-1→(O‘ω))
64	61, 63	sylibr 237	. . . . . . . . . . . . . . . 16 ⊢ (ω ∈ V → ω ≼ (O‘ω))
65		infinfg 10657	. . . . . . . . . . . . . . . . 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 2880	. . . . . . . . . . . . 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 7873	. 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 1569  ∃wex 1811   ∈ wcel 2144  ∀wral 3076  Vcvv 3450   ⊆ wss 3896  𝒫 cpw 4556  ∪ cuni 4866   class class class wbr 5102  ^◡ccnv 5650  dom cdm 5651   ↾ cres 5653   “ cima 5654  Ord word 6364  Oncon0 6365  Fun wfun 6537  ⟶wf 6539  –1-1→wf1 6540  –1-1-onto→wf1o 6542  ‘cfv 6543  ωcom 7881   ≼ cdom 8978  Fincfn 8980  Nocsur 28006  bdaycbday 28008  Mcmade 28217  Ocold 28218  ℕ_0scn0s 28707

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755  ax-ac2 10548

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-ot 4858  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  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 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7882  df-1st 8005  df-2nd 8006  df-frecs 8306  df-wrecs 8337  df-recs 8386  df-rdg 8425  df-1o 8483  df-2o 8484  df-nadd 8682  df-er 8724  df-en 8981  df-dom 8982  df-sdom 8983  df-fin 8984  df-card 10027  df-ac 10202  df-no 28009  df-lts 28010  df-bday 28011  df-les 28111  df-slts 28153  df-cuts 28155  df-0s 28202  df-1s 28203  df-made 28222  df-old 28223  df-left 28225  df-right 28226  df-norec 28333  df-norec2 28344  df-adds 28355  df-negs 28416  df-subs 28417  df-ons 28647  df-n0s 28709

This theorem is used by: (None)