Theorem oldfib 28570

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 28107	. 2 ⊢ (𝐴 ∈ ω → ( O ‘𝐴) ∈ Fin)
2		fveq2 6881	. . . . 5 ⊢ (𝑥 = 𝑦 → ( O ‘𝑥) = ( O ‘𝑦))
3	2	eleq1d 2848	. . . 4 ⊢ (𝑥 = 𝑦 → (( O ‘𝑥) ∈ Fin ↔ ( O ‘𝑦) ∈ Fin))
4		eleq1 2851	. . . 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 6881	. . . . 5 ⊢ (𝑥 = 𝐴 → ( O ‘𝑥) = ( O ‘𝐴))
7	6	eleq1d 2848	. . . 4 ⊢ (𝑥 = 𝐴 → (( O ‘𝑥) ∈ Fin ↔ ( O ‘𝐴) ∈ Fin))
8		eleq1 2851	. . . 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 28027	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → ( O ‘𝑥) = ∪ ( M “ 𝑥))
11	10	eleq1d 2848	. . . . . . . . . 10 ⊢ (𝑥 ∈ On → (( O ‘𝑥) ∈ Fin ↔ ∪ ( M “ 𝑥) ∈ Fin))
12	11	biimpa 481	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ ( O ‘𝑥) ∈ Fin) → ∪ ( M “ 𝑥) ∈ Fin)
13		unifi3 9315	. . . . . . . . 9 ⊢ (∪ ( M “ 𝑥) ∈ Fin → ( M “ 𝑥) ⊆ Fin)
14	12, 13	syl 18	. . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ ( O ‘𝑥) ∈ Fin) → ( M “ 𝑥) ⊆ Fin)
15		madef 28029	. . . . . . . . . . 11 ⊢ M :On⟶𝒫 No
16		ffun 6708	. . . . . . . . . . 11 ⊢ ( M :On⟶𝒫 No → Fun M )
17	15, 16	ax-mp 5	. . . . . . . . . 10 ⊢ Fun M
18		onss 7780	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → 𝑥 ⊆ On)
19	15	fdmi 6717	. . . . . . . . . . 11 ⊢ dom M = On
20	18, 19	sseqtrrdi 3978	. . . . . . . . . 10 ⊢ (𝑥 ∈ On → 𝑥 ⊆ dom M )
21		funimass4 6945	. . . . . . . . . 10 ⊢ ((Fun M ∧ 𝑥 ⊆ dom M ) → (( M “ 𝑥) ⊆ Fin ↔ ∀𝑦 ∈ 𝑥 ( M ‘𝑦) ∈ Fin))
22	17, 20, 21	sylancr 598	. . . . . . . . 9 ⊢ (𝑥 ∈ On → (( M “ 𝑥) ⊆ Fin ↔ ∀𝑦 ∈ 𝑥 ( M ‘𝑦) ∈ Fin))
23	22	adantr 485	. . . . . . . 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 28060	. . . . . . . . 9 ⊢ ( O ‘𝑦) ⊆ ( M ‘𝑦)
26		ssfi 9153	. . . . . . . . 9 ⊢ ((( M ‘𝑦) ∈ Fin ∧ ( O ‘𝑦) ⊆ ( M ‘𝑦)) → ( O ‘𝑦) ∈ Fin)
27	25, 26	mpan2 703	. . . . . . . 8 ⊢ (( M ‘𝑦) ∈ Fin → ( O ‘𝑦) ∈ Fin)
28	27	ralimi 3102	. . . . . . 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 3125	. . . . . . . . . 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 412	. . . . . . . . . . . 12 ⊢ (((( O ‘𝑦) ∈ Fin → 𝑦 ∈ ω) ∧ ( O ‘𝑦) ∈ Fin) → 𝑦 ∈ ω)
34	33	ralimi 3102	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ 𝑥 ((( O ‘𝑦) ∈ Fin → 𝑦 ∈ ω) ∧ ( O ‘𝑦) ∈ Fin) → ∀𝑦 ∈ 𝑥 𝑦 ∈ ω)
35		dfss3 3926	. . . . . . . . . . 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 6370	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → Ord 𝑥)
39		ordom 7868	. . . . . . . . . . . . 13 ⊢ Ord ω
40		ordsseleq 6390	. . . . . . . . . . . . 13 ⊢ ((Ord 𝑥 ∧ Ord ω) → (𝑥 ⊆ ω ↔ (𝑥 ∈ ω ∨ 𝑥 = ω)))
41	39, 40	mpan2 703	. . . . . . . . . . . 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 485	. . . . . . . . . 10 ⊢ ((𝑥 ∈ On ∧ ( O ‘𝑥) ∈ Fin) → (𝑥 ⊆ ω ↔ (𝑥 ∈ ω ∨ 𝑥 = ω)))
44		fveq2 6881	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ω → ( O ‘𝑥) = ( O ‘ω))
45		eqvisset 3475	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = ω → ω ∈ V)
46		bdayfun 27940	. . . . . . . . . . . . . . . . . . . 20 ⊢ Fun bday
47		n0sexg 28509	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ω ∈ V → ℕ0s ∈ V)
48		resfunexg 7213	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((Fun bday ∧ ℕ0s ∈ V) → ( bday ↾ ℕ0s) ∈ V)
49	46, 47, 48	sylancr 598	. . . . . . . . . . . . . . . . . . 19 ⊢ (ω ∈ V → ( bday ↾ ℕ0s) ∈ V)
50		cnvexg 7917	. . . . . . . . . . . . . . . . . . 19 ⊢ (( bday ↾ ℕ0s) ∈ V → ◡( bday ↾ ℕ0s) ∈ V)
51	49, 50	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (ω ∈ V → ◡( bday ↾ ℕ0s) ∈ V)
52		bdayn0sf1o 28563	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ( bday ↾ ℕ0s):ℕ0s–1-1-onto→ω
53	52	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ω ∈ V → ( bday ↾ ℕ0s):ℕ0s–1-1-onto→ω)
54		f1ocnv 6833	. . . . . . . . . . . . . . . . . . . 20 ⊢ (( bday ↾ ℕ0s):ℕ0s–1-1-onto→ω → ◡( bday ↾ ℕ0s):ω–1-1-onto→ℕ0s)
55		f1of1 6819	. . . . . . . . . . . . . . . . . . . 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 28546	. . . . . . . . . . . . . . . . . . 19 ⊢ (ω ∈ V → ℕ0s ⊆ ( O ‘ω))
58		f1ss 6781	. . . . . . . . . . . . . . . . . . 19 ⊢ ((◡( bday ↾ ℕ0s):ω–1-1→ℕ0s ∧ ℕ0s ⊆ ( O ‘ω)) → ◡( bday ↾ ℕ0s):ω–1-1→( O ‘ω))
59	56, 57, 58	syl2anc 595	. . . . . . . . . . . . . . . . . 18 ⊢ (ω ∈ V → ◡( bday ↾ ℕ0s):ω–1-1→( O ‘ω))
60		f1eq1 6769	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = ◡( bday ↾ ℕ0s) → (𝑓:ω–1-1→( O ‘ω) ↔ ◡( bday ↾ ℕ0s):ω–1-1→( O ‘ω)))
61	51, 59, 60	spcedv 3557	. . . . . . . . . . . . . . . . 17 ⊢ (ω ∈ V → ∃𝑓 𝑓:ω–1-1→( O ‘ω))
62		fvex 6894	. . . . . . . . . . . . . . . . . 18 ⊢ ( O ‘ω) ∈ V
63	62	brdom 8953	. . . . . . . . . . . . . . . . 17 ⊢ (ω ≼ ( O ‘ω) ↔ ∃𝑓 𝑓:ω–1-1→( O ‘ω))
64	61, 63	sylibr 237	. . . . . . . . . . . . . . . 16 ⊢ (ω ∈ V → ω ≼ ( O ‘ω))
65		infinfg 10545	. . . . . . . . . . . . . . . . 17 ⊢ ((ω ∈ V ∧ ( O ‘ω) ∈ V) → (¬ ( O ‘ω) ∈ Fin ↔ ω ≼ ( O ‘ω)))
66	62, 65	mpan2 703	. . . . . . . . . . . . . . . 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 2883	. . . . . . . . . . . . 13 ⊢ (𝑥 = ω → ¬ ( O ‘𝑥) ∈ Fin)
70	69	con2i 140	. . . . . . . . . . . 12 ⊢ (( O ‘𝑥) ∈ Fin → ¬ 𝑥 = ω)
71	70	adantl 486	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ On ∧ ( O ‘𝑥) ∈ Fin) → ¬ 𝑥 = ω)
72		orel2 903	. . . . . . . . . . 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 420	. . . . . . 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 7850	. 2 ⊢ (𝐴 ∈ On → (( O ‘𝐴) ∈ Fin → 𝐴 ∈ ω))
82	1, 81	impbid2 229	1 ⊢ (𝐴 ∈ On → (𝐴 ∈ ω ↔ ( O ‘𝐴) ∈ Fin))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∧ w3a 1103   = wceq 1570  ∃wex 1809   ∈ wcel 2143  ∀wral 3079  Vcvv 3455   ⊆ wss 3905  𝒫 cpw 4562  ∪ cuni 4872   class class class wbr 5109  ^◡ccnv 5660  dom cdm 5661   ↾ cres 5663   “ cima 5664  Ord word 6359  Oncon0 6360  Fun wfun 6530  ⟶wf 6532  –1-1→wf1 6533  –1-1-onto→wf1o 6535  ‘cfv 6536  ωcom 7858   ≼ cdom 8937  Fincfn 8939   No csur 27804   bday cbday 27806   M cmade 28015   O cold 28016  ℕ_0scn0s 28505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-ac2 10442

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-ot 4598  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-2o 8450  df-nadd 8648  df-er 8690  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-card 9921  df-acn 9924  df-ac 10096  df-no 27807  df-lts 27808  df-bday 27809  df-les 27909  df-slts 27951  df-cuts 27953  df-0s 28000  df-1s 28001  df-made 28020  df-old 28021  df-left 28023  df-right 28024  df-norec 28131  df-norec2 28142  df-adds 28153  df-negs 28214  df-subs 28215  df-ons 28445  df-n0s 28507

This theorem is referenced by: (None)