Theorem oldfib 28385

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 27922	. 2 ⊢ (𝐴 ∈ ω → ( O ‘𝐴) ∈ Fin)
2		fveq2 6842	. . . . 5 ⊢ (𝑥 = 𝑦 → ( O ‘𝑥) = ( O ‘𝑦))
3	2	eleq1d 2822	. . . 4 ⊢ (𝑥 = 𝑦 → (( O ‘𝑥) ∈ Fin ↔ ( O ‘𝑦) ∈ Fin))
4		eleq1 2825	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ ω ↔ 𝑦 ∈ ω))
5	3, 4	imbi12d 344	. . 3 ⊢ (𝑥 = 𝑦 → ((( O ‘𝑥) ∈ Fin → 𝑥 ∈ ω) ↔ (( O ‘𝑦) ∈ Fin → 𝑦 ∈ ω)))
6		fveq2 6842	. . . . 5 ⊢ (𝑥 = 𝐴 → ( O ‘𝑥) = ( O ‘𝐴))
7	6	eleq1d 2822	. . . 4 ⊢ (𝑥 = 𝐴 → (( O ‘𝑥) ∈ Fin ↔ ( O ‘𝐴) ∈ Fin))
8		eleq1 2825	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ ω ↔ 𝐴 ∈ ω))
9	7, 8	imbi12d 344	. . 3 ⊢ (𝑥 = 𝐴 → ((( O ‘𝑥) ∈ Fin → 𝑥 ∈ ω) ↔ (( O ‘𝐴) ∈ Fin → 𝐴 ∈ ω)))
10		oldval 27842	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → ( O ‘𝑥) = ∪ ( M “ 𝑥))
11	10	eleq1d 2822	. . . . . . . . . 10 ⊢ (𝑥 ∈ On → (( O ‘𝑥) ∈ Fin ↔ ∪ ( M “ 𝑥) ∈ Fin))
12	11	biimpa 476	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ ( O ‘𝑥) ∈ Fin) → ∪ ( M “ 𝑥) ∈ Fin)
13		unifi3 9274	. . . . . . . . 9 ⊢ (∪ ( M “ 𝑥) ∈ Fin → ( M “ 𝑥) ⊆ Fin)
14	12, 13	syl 17	. . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ ( O ‘𝑥) ∈ Fin) → ( M “ 𝑥) ⊆ Fin)
15		madef 27844	. . . . . . . . . . 11 ⊢ M :On⟶𝒫 No
16		ffun 6673	. . . . . . . . . . 11 ⊢ ( M :On⟶𝒫 No → Fun M )
17	15, 16	ax-mp 5	. . . . . . . . . 10 ⊢ Fun M
18		onss 7740	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → 𝑥 ⊆ On)
19	15	fdmi 6681	. . . . . . . . . . 11 ⊢ dom M = On
20	18, 19	sseqtrrdi 3977	. . . . . . . . . 10 ⊢ (𝑥 ∈ On → 𝑥 ⊆ dom M )
21		funimass4 6906	. . . . . . . . . 10 ⊢ ((Fun M ∧ 𝑥 ⊆ dom M ) → (( M “ 𝑥) ⊆ Fin ↔ ∀𝑦 ∈ 𝑥 ( M ‘𝑦) ∈ Fin))
22	17, 20, 21	sylancr 588	. . . . . . . . 9 ⊢ (𝑥 ∈ On → (( M “ 𝑥) ⊆ Fin ↔ ∀𝑦 ∈ 𝑥 ( M ‘𝑦) ∈ Fin))
23	22	adantr 480	. . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ ( O ‘𝑥) ∈ Fin) → (( M “ 𝑥) ⊆ Fin ↔ ∀𝑦 ∈ 𝑥 ( M ‘𝑦) ∈ Fin))
24	14, 23	mpbid 232	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ ( O ‘𝑥) ∈ Fin) → ∀𝑦 ∈ 𝑥 ( M ‘𝑦) ∈ Fin)
25		oldssmade 27875	. . . . . . . . 9 ⊢ ( O ‘𝑦) ⊆ ( M ‘𝑦)
26		ssfi 9109	. . . . . . . . 9 ⊢ ((( M ‘𝑦) ∈ Fin ∧ ( O ‘𝑦) ⊆ ( M ‘𝑦)) → ( O ‘𝑦) ∈ Fin)
27	25, 26	mpan2 692	. . . . . . . 8 ⊢ (( M ‘𝑦) ∈ Fin → ( O ‘𝑦) ∈ Fin)
28	27	ralimi 3075	. . . . . . 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 1132	. . . . 5 ⊢ ((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (( O ‘𝑦) ∈ Fin → 𝑦 ∈ ω) ∧ ( O ‘𝑥) ∈ Fin) → ∀𝑦 ∈ 𝑥 ( O ‘𝑦) ∈ Fin)
31		r19.26 3098	. . . . . . . . . 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 407	. . . . . . . . . . . 12 ⊢ (((( O ‘𝑦) ∈ Fin → 𝑦 ∈ ω) ∧ ( O ‘𝑦) ∈ Fin) → 𝑦 ∈ ω)
34	33	ralimi 3075	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ 𝑥 ((( O ‘𝑦) ∈ Fin → 𝑦 ∈ ω) ∧ ( O ‘𝑦) ∈ Fin) → ∀𝑦 ∈ 𝑥 𝑦 ∈ ω)
35		dfss3 3924	. . . . . . . . . . 11 ⊢ (𝑥 ⊆ ω ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ ω)
36	34, 35	sylibr 234	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝑥 ((( O ‘𝑦) ∈ Fin → 𝑦 ∈ ω) ∧ ( O ‘𝑦) ∈ Fin) → 𝑥 ⊆ ω)
37	31, 36	sylbir 235	. . . . . . . . 9 ⊢ ((∀𝑦 ∈ 𝑥 (( O ‘𝑦) ∈ Fin → 𝑦 ∈ ω) ∧ ∀𝑦 ∈ 𝑥 ( O ‘𝑦) ∈ Fin) → 𝑥 ⊆ ω)
38		eloni 6335	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → Ord 𝑥)
39		ordom 7828	. . . . . . . . . . . . 13 ⊢ Ord ω
40		ordsseleq 6354	. . . . . . . . . . . . 13 ⊢ ((Ord 𝑥 ∧ Ord ω) → (𝑥 ⊆ ω ↔ (𝑥 ∈ ω ∨ 𝑥 = ω)))
41	39, 40	mpan2 692	. . . . . . . . . . . 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 480	. . . . . . . . . 10 ⊢ ((𝑥 ∈ On ∧ ( O ‘𝑥) ∈ Fin) → (𝑥 ⊆ ω ↔ (𝑥 ∈ ω ∨ 𝑥 = ω)))
44		fveq2 6842	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ω → ( O ‘𝑥) = ( O ‘ω))
45		eqvisset 3462	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = ω → ω ∈ V)
46		bdayfun 27756	. . . . . . . . . . . . . . . . . . . 20 ⊢ Fun bday
47		n0sexg 28324	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ω ∈ V → ℕ0s ∈ V)
48		resfunexg 7171	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((Fun bday ∧ ℕ0s ∈ V) → ( bday ↾ ℕ0s) ∈ V)
49	46, 47, 48	sylancr 588	. . . . . . . . . . . . . . . . . . 19 ⊢ (ω ∈ V → ( bday ↾ ℕ0s) ∈ V)
50		cnvexg 7876	. . . . . . . . . . . . . . . . . . 19 ⊢ (( bday ↾ ℕ0s) ∈ V → ◡( bday ↾ ℕ0s) ∈ V)
51	49, 50	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (ω ∈ V → ◡( bday ↾ ℕ0s) ∈ V)
52		bdayn0sf1o 28378	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ( bday ↾ ℕ0s):ℕ0s–1-1-onto→ω
53	52	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ω ∈ V → ( bday ↾ ℕ0s):ℕ0s–1-1-onto→ω)
54		f1ocnv 6794	. . . . . . . . . . . . . . . . . . . 20 ⊢ (( bday ↾ ℕ0s):ℕ0s–1-1-onto→ω → ◡( bday ↾ ℕ0s):ω–1-1-onto→ℕ0s)
55		f1of1 6781	. . . . . . . . . . . . . . . . . . . 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 28361	. . . . . . . . . . . . . . . . . . 19 ⊢ (ω ∈ V → ℕ0s ⊆ ( O ‘ω))
58		f1ss 6743	. . . . . . . . . . . . . . . . . . 19 ⊢ ((◡( bday ↾ ℕ0s):ω–1-1→ℕ0s ∧ ℕ0s ⊆ ( O ‘ω)) → ◡( bday ↾ ℕ0s):ω–1-1→( O ‘ω))
59	56, 57, 58	syl2anc 585	. . . . . . . . . . . . . . . . . 18 ⊢ (ω ∈ V → ◡( bday ↾ ℕ0s):ω–1-1→( O ‘ω))
60		f1eq1 6733	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = ◡( bday ↾ ℕ0s) → (𝑓:ω–1-1→( O ‘ω) ↔ ◡( bday ↾ ℕ0s):ω–1-1→( O ‘ω)))
61	51, 59, 60	spcedv 3554	. . . . . . . . . . . . . . . . 17 ⊢ (ω ∈ V → ∃𝑓 𝑓:ω–1-1→( O ‘ω))
62		fvex 6855	. . . . . . . . . . . . . . . . . 18 ⊢ ( O ‘ω) ∈ V
63	62	brdom 8909	. . . . . . . . . . . . . . . . 17 ⊢ (ω ≼ ( O ‘ω) ↔ ∃𝑓 𝑓:ω–1-1→( O ‘ω))
64	61, 63	sylibr 234	. . . . . . . . . . . . . . . 16 ⊢ (ω ∈ V → ω ≼ ( O ‘ω))
65		infinfg 10488	. . . . . . . . . . . . . . . . 17 ⊢ ((ω ∈ V ∧ ( O ‘ω) ∈ V) → (¬ ( O ‘ω) ∈ Fin ↔ ω ≼ ( O ‘ω)))
66	62, 65	mpan2 692	. . . . . . . . . . . . . . . 16 ⊢ (ω ∈ V → (¬ ( O ‘ω) ∈ Fin ↔ ω ≼ ( O ‘ω)))
67	64, 66	mpbird 257	. . . . . . . . . . . . . . 15 ⊢ (ω ∈ V → ¬ ( O ‘ω) ∈ Fin)
68	45, 67	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ω → ¬ ( O ‘ω) ∈ Fin)
69	44, 68	eqneltrd 2857	. . . . . . . . . . . . 13 ⊢ (𝑥 = ω → ¬ ( O ‘𝑥) ∈ Fin)
70	69	con2i 139	. . . . . . . . . . . 12 ⊢ (( O ‘𝑥) ∈ Fin → ¬ 𝑥 = ω)
71	70	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ On ∧ ( O ‘𝑥) ∈ Fin) → ¬ 𝑥 = ω)
72		orel2 891	. . . . . . . . . . 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 240	. . . . . . . . 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 415	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ ( O ‘𝑥) ∈ Fin) → (∀𝑦 ∈ 𝑥 (( O ‘𝑦) ∈ Fin → 𝑦 ∈ ω) → (∀𝑦 ∈ 𝑥 ( O ‘𝑦) ∈ Fin → 𝑥 ∈ ω)))
77	76	3impia 1118	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ ( O ‘𝑥) ∈ Fin ∧ ∀𝑦 ∈ 𝑥 (( O ‘𝑦) ∈ Fin → 𝑦 ∈ ω)) → (∀𝑦 ∈ 𝑥 ( O ‘𝑦) ∈ Fin → 𝑥 ∈ ω))
78	77	3com23 1127	. . . . 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 1120	. . 3 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 (( O ‘𝑦) ∈ Fin → 𝑦 ∈ ω) → (( O ‘𝑥) ∈ Fin → 𝑥 ∈ ω)))
81	5, 9, 80	tfis3 7810	. 2 ⊢ (𝐴 ∈ On → (( O ‘𝐴) ∈ Fin → 𝐴 ∈ ω))
82	1, 81	impbid2 226	1 ⊢ (𝐴 ∈ On → (𝐴 ∈ ω ↔ ( O ‘𝐴) ∈ Fin))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∀wral 3052  Vcvv 3442   ⊆ wss 3903  𝒫 cpw 4556  ∪ cuni 4865   class class class wbr 5100  ^◡ccnv 5631  dom cdm 5632   ↾ cres 5634   “ cima 5635  Ord word 6324  Oncon0 6325  Fun wfun 6494  ⟶wf 6496  –1-1→wf1 6497  –1-1-onto→wf1o 6499  ‘cfv 6500  ωcom 7818   ≼ cdom 8893  Fincfn 8895   No csur 27619   bday cbday 27621   M cmade 27830   O cold 27831  ℕ_0scn0s 28320

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-ac2 10385

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-ot 4591  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-2o 8408  df-nadd 8604  df-er 8645  df-map 8777  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-card 9863  df-acn 9866  df-ac 10038  df-no 27622  df-lts 27623  df-bday 27624  df-les 27725  df-slts 27766  df-cuts 27768  df-0s 27815  df-1s 27816  df-made 27835  df-old 27836  df-left 27838  df-right 27839  df-norec 27946  df-norec2 27957  df-adds 27968  df-negs 28029  df-subs 28030  df-ons 28260  df-n0s 28322

This theorem is referenced by: (None)