Theorem onsfi 28769

Description: A surreal ordinal with a finite birthday is a non-negative surreal integer. (Contributed by Scott Fenton, 4-Nov-2025.)

Assertion

Ref	Expression
onsfi	⊢ ((𝐴 ∈ Ons ∧ (bday‘𝐴) ∈ ω) → 𝐴 ∈ ℕ0s)

Proof of Theorem onsfi

Dummy variables 𝑎 𝑏 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		risset 3241	. . 3 ⊢ ((bday‘𝐴) ∈ ω ↔ ∃𝑥 ∈ ω 𝑥 = (bday‘𝐴))
2		eqeq1 2767	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → (𝑦 = (bday‘𝑎) ↔ 𝑧 = (bday‘𝑎)))
3	2	imbi1d 344	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → ((𝑦 = (bday‘𝑎) → 𝑎 ∈ ℕ0s) ↔ (𝑧 = (bday‘𝑎) → 𝑎 ∈ ℕ0s)))
4	3	ralbidv 3189	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → (∀𝑎 ∈ Ons (𝑦 = (bday‘𝑎) → 𝑎 ∈ ℕ0s) ↔ ∀𝑎 ∈ Ons (𝑧 = (bday‘𝑎) → 𝑎 ∈ ℕ0s)))
5		fveq2 6905	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑏 → (bday‘𝑎) = (bday‘𝑏))
6	5	eqeq2d 2774	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → (𝑧 = (bday‘𝑎) ↔ 𝑧 = (bday‘𝑏)))
7		eleq1 2852	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → (𝑎 ∈ ℕ0s ↔ 𝑏 ∈ ℕ0s))
8	6, 7	imbi12d 347	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → ((𝑧 = (bday‘𝑎) → 𝑎 ∈ ℕ0s) ↔ (𝑧 = (bday‘𝑏) → 𝑏 ∈ ℕ0s)))
9	8	cbvralvw 3244	. . . . . . . 8 ⊢ (∀𝑎 ∈ Ons (𝑧 = (bday‘𝑎) → 𝑎 ∈ ℕ0s) ↔ ∀𝑏 ∈ Ons (𝑧 = (bday‘𝑏) → 𝑏 ∈ ℕ0s))
10	4, 9	bitrdi 290	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (∀𝑎 ∈ Ons (𝑦 = (bday‘𝑎) → 𝑎 ∈ ℕ0s) ↔ ∀𝑏 ∈ Ons (𝑧 = (bday‘𝑏) → 𝑏 ∈ ℕ0s)))
11		eqeq1 2767	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝑦 = (bday‘𝑎) ↔ 𝑥 = (bday‘𝑎)))
12	11	imbi1d 344	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → ((𝑦 = (bday‘𝑎) → 𝑎 ∈ ℕ0s) ↔ (𝑥 = (bday‘𝑎) → 𝑎 ∈ ℕ0s)))
13	12	ralbidv 3189	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (∀𝑎 ∈ Ons (𝑦 = (bday‘𝑎) → 𝑎 ∈ ℕ0s) ↔ ∀𝑎 ∈ Ons (𝑥 = (bday‘𝑎) → 𝑎 ∈ ℕ0s)))
14		oncutlt 28677	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ Ons → 𝑎 = ({𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} |s ∅))
15	14	3ad2ant3 1153	. . . . . . . . . . . 12 ⊢ (((bday‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons ((bday‘𝑏) ∈ (bday‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) → 𝑎 = ({𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} |s ∅))
16		onssno 28667	. . . . . . . . . . . . . . . . . . 19 ⊢ Ons ⊆ No
17		simp13 1224	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((bday‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons ((bday‘𝑏) ∈ (bday‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) ∧ 𝑥 ∈ Ons ∧ 𝑥 <s 𝑎) → 𝑎 ∈ Ons)
18	16, 17	sselid 3911	. . . . . . . . . . . . . . . . . 18 ⊢ ((((bday‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons ((bday‘𝑏) ∈ (bday‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) ∧ 𝑥 ∈ Ons ∧ 𝑥 <s 𝑎) → 𝑎 ∈ No)
19		ltonold 28674	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 ∈ No → {𝑏 ∈ Ons ∣ 𝑏 <s 𝑎} ⊆ (O‘(bday‘𝑎)))
20	18, 19	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((((bday‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons ((bday‘𝑏) ∈ (bday‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) ∧ 𝑥 ∈ Ons ∧ 𝑥 <s 𝑎) → {𝑏 ∈ Ons ∣ 𝑏 <s 𝑎} ⊆ (O‘(bday‘𝑎)))
21		breq1 5109	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑥 → (𝑏 <s 𝑎 ↔ 𝑥 <s 𝑎))
22		simp2 1155	. . . . . . . . . . . . . . . . . 18 ⊢ ((((bday‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons ((bday‘𝑏) ∈ (bday‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) ∧ 𝑥 ∈ Ons ∧ 𝑥 <s 𝑎) → 𝑥 ∈ Ons)
23		simp3 1156	. . . . . . . . . . . . . . . . . 18 ⊢ ((((bday‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons ((bday‘𝑏) ∈ (bday‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) ∧ 𝑥 ∈ Ons ∧ 𝑥 <s 𝑎) → 𝑥 <s 𝑎)
24	21, 22, 23	elrabd 3650	. . . . . . . . . . . . . . . . 17 ⊢ ((((bday‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons ((bday‘𝑏) ∈ (bday‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) ∧ 𝑥 ∈ Ons ∧ 𝑥 <s 𝑎) → 𝑥 ∈ {𝑏 ∈ Ons ∣ 𝑏 <s 𝑎})
25	20, 24	sseldd 3914	. . . . . . . . . . . . . . . 16 ⊢ ((((bday‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons ((bday‘𝑏) ∈ (bday‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) ∧ 𝑥 ∈ Ons ∧ 𝑥 <s 𝑎) → 𝑥 ∈ (O‘(bday‘𝑎)))
26		bdayon 28165	. . . . . . . . . . . . . . . . 17 ⊢ (bday‘𝑎) ∈ On
27	16, 22	sselid 3911	. . . . . . . . . . . . . . . . 17 ⊢ ((((bday‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons ((bday‘𝑏) ∈ (bday‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) ∧ 𝑥 ∈ Ons ∧ 𝑥 <s 𝑎) → 𝑥 ∈ No)
28		oldbday 28314	. . . . . . . . . . . . . . . . 17 ⊢ (((bday‘𝑎) ∈ On ∧ 𝑥 ∈ No) → (𝑥 ∈ (O‘(bday‘𝑎)) ↔ (bday‘𝑥) ∈ (bday‘𝑎)))
29	26, 27, 28	sylancr 599	. . . . . . . . . . . . . . . 16 ⊢ ((((bday‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons ((bday‘𝑏) ∈ (bday‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) ∧ 𝑥 ∈ Ons ∧ 𝑥 <s 𝑎) → (𝑥 ∈ (O‘(bday‘𝑎)) ↔ (bday‘𝑥) ∈ (bday‘𝑎)))
30	25, 29	mpbid 235	. . . . . . . . . . . . . . 15 ⊢ ((((bday‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons ((bday‘𝑏) ∈ (bday‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) ∧ 𝑥 ∈ Ons ∧ 𝑥 <s 𝑎) → (bday‘𝑥) ∈ (bday‘𝑎))
31		fveq2 6905	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑥 → (bday‘𝑏) = (bday‘𝑥))
32	31	eleq1d 2849	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑥 → ((bday‘𝑏) ∈ (bday‘𝑎) ↔ (bday‘𝑥) ∈ (bday‘𝑎)))
33		eleq1 2852	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑥 → (𝑏 ∈ ℕ0s ↔ 𝑥 ∈ ℕ0s))
34	32, 33	imbi12d 347	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 𝑥 → (((bday‘𝑏) ∈ (bday‘𝑎) → 𝑏 ∈ ℕ0s) ↔ ((bday‘𝑥) ∈ (bday‘𝑎) → 𝑥 ∈ ℕ0s)))
35		simp12 1223	. . . . . . . . . . . . . . . 16 ⊢ ((((bday‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons ((bday‘𝑏) ∈ (bday‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) ∧ 𝑥 ∈ Ons ∧ 𝑥 <s 𝑎) → ∀𝑏 ∈ Ons ((bday‘𝑏) ∈ (bday‘𝑎) → 𝑏 ∈ ℕ0s))
36	34, 35, 22	rspcdva 3581	. . . . . . . . . . . . . . 15 ⊢ ((((bday‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons ((bday‘𝑏) ∈ (bday‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) ∧ 𝑥 ∈ Ons ∧ 𝑥 <s 𝑎) → ((bday‘𝑥) ∈ (bday‘𝑎) → 𝑥 ∈ ℕ0s))
37	30, 36	mpd 16	. . . . . . . . . . . . . 14 ⊢ ((((bday‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons ((bday‘𝑏) ∈ (bday‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) ∧ 𝑥 ∈ Ons ∧ 𝑥 <s 𝑎) → 𝑥 ∈ ℕ0s)
38	37	rabssdv 4002	. . . . . . . . . . . . 13 ⊢ (((bday‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons ((bday‘𝑏) ∈ (bday‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) → {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} ⊆ ℕ0s)
39		oldfi 28327	. . . . . . . . . . . . . . 15 ⊢ ((bday‘𝑎) ∈ ω → (O‘(bday‘𝑎)) ∈ Fin)
40	39	3ad2ant1 1151	. . . . . . . . . . . . . 14 ⊢ (((bday‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons ((bday‘𝑏) ∈ (bday‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) → (O‘(bday‘𝑎)) ∈ Fin)
41		onno 28668	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ Ons → 𝑎 ∈ No)
42	41	3ad2ant3 1153	. . . . . . . . . . . . . . 15 ⊢ (((bday‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons ((bday‘𝑏) ∈ (bday‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) → 𝑎 ∈ No)
43		ltonold 28674	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ No → {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} ⊆ (O‘(bday‘𝑎)))
44	42, 43	syl 18	. . . . . . . . . . . . . 14 ⊢ (((bday‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons ((bday‘𝑏) ∈ (bday‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) → {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} ⊆ (O‘(bday‘𝑎)))
45	40, 44	ssfid 9285	. . . . . . . . . . . . 13 ⊢ (((bday‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons ((bday‘𝑏) ∈ (bday‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) → {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} ∈ Fin)
46		n0fincut 28768	. . . . . . . . . . . . 13 ⊢ (({𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} ⊆ ℕ0s ∧ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} ∈ Fin) → ({𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} |s ∅) ∈ ℕ0s)
47	38, 45, 46	syl2anc 596	. . . . . . . . . . . 12 ⊢ (((bday‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons ((bday‘𝑏) ∈ (bday‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) → ({𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} |s ∅) ∈ ℕ0s)
48	15, 47	eqeltrd 2864	. . . . . . . . . . 11 ⊢ (((bday‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons ((bday‘𝑏) ∈ (bday‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) → 𝑎 ∈ ℕ0s)
49	48	3exp 1137	. . . . . . . . . 10 ⊢ ((bday‘𝑎) ∈ ω → (∀𝑏 ∈ Ons ((bday‘𝑏) ∈ (bday‘𝑎) → 𝑏 ∈ ℕ0s) → (𝑎 ∈ Ons → 𝑎 ∈ ℕ0s)))
50		eleq1 2852	. . . . . . . . . . 11 ⊢ (𝑦 = (bday‘𝑎) → (𝑦 ∈ ω ↔ (bday‘𝑎) ∈ ω))
51		raleq 3320	. . . . . . . . . . . . 13 ⊢ (𝑦 = (bday‘𝑎) → (∀𝑧 ∈ 𝑦 ∀𝑏 ∈ Ons (𝑧 = (bday‘𝑏) → 𝑏 ∈ ℕ0s) ↔ ∀𝑧 ∈ (bday‘𝑎)∀𝑏 ∈ Ons (𝑧 = (bday‘𝑏) → 𝑏 ∈ ℕ0s)))
52		ralcom 3294	. . . . . . . . . . . . . 14 ⊢ (∀𝑧 ∈ (bday‘𝑎)∀𝑏 ∈ Ons (𝑧 = (bday‘𝑏) → 𝑏 ∈ ℕ0s) ↔ ∀𝑏 ∈ Ons ∀𝑧 ∈ (bday‘𝑎)(𝑧 = (bday‘𝑏) → 𝑏 ∈ ℕ0s))
53		df-ral 3081	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑧 ∈ (bday‘𝑎)(𝑧 = (bday‘𝑏) → 𝑏 ∈ ℕ0s) ↔ ∀𝑧(𝑧 ∈ (bday‘𝑎) → (𝑧 = (bday‘𝑏) → 𝑏 ∈ ℕ0s)))
54		bi2.04 392	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ (bday‘𝑎) → (𝑧 = (bday‘𝑏) → 𝑏 ∈ ℕ0s)) ↔ (𝑧 = (bday‘𝑏) → (𝑧 ∈ (bday‘𝑎) → 𝑏 ∈ ℕ0s)))
55	54	albii 1852	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑧(𝑧 ∈ (bday‘𝑎) → (𝑧 = (bday‘𝑏) → 𝑏 ∈ ℕ0s)) ↔ ∀𝑧(𝑧 = (bday‘𝑏) → (𝑧 ∈ (bday‘𝑎) → 𝑏 ∈ ℕ0s)))
56		fvex 6918	. . . . . . . . . . . . . . . . 17 ⊢ (bday‘𝑏) ∈ V
57		eleq1 2852	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = (bday‘𝑏) → (𝑧 ∈ (bday‘𝑎) ↔ (bday‘𝑏) ∈ (bday‘𝑎)))
58	57	imbi1d 344	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (bday‘𝑏) → ((𝑧 ∈ (bday‘𝑎) → 𝑏 ∈ ℕ0s) ↔ ((bday‘𝑏) ∈ (bday‘𝑎) → 𝑏 ∈ ℕ0s)))
59	56, 58	ceqsalv 3493	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑧(𝑧 = (bday‘𝑏) → (𝑧 ∈ (bday‘𝑎) → 𝑏 ∈ ℕ0s)) ↔ ((bday‘𝑏) ∈ (bday‘𝑎) → 𝑏 ∈ ℕ0s))
60	53, 55, 59	3bitri 300	. . . . . . . . . . . . . . 15 ⊢ (∀𝑧 ∈ (bday‘𝑎)(𝑧 = (bday‘𝑏) → 𝑏 ∈ ℕ0s) ↔ ((bday‘𝑏) ∈ (bday‘𝑎) → 𝑏 ∈ ℕ0s))
61	60	ralbii 3112	. . . . . . . . . . . . . 14 ⊢ (∀𝑏 ∈ Ons ∀𝑧 ∈ (bday‘𝑎)(𝑧 = (bday‘𝑏) → 𝑏 ∈ ℕ0s) ↔ ∀𝑏 ∈ Ons ((bday‘𝑏) ∈ (bday‘𝑎) → 𝑏 ∈ ℕ0s))
62	52, 61	bitri 278	. . . . . . . . . . . . 13 ⊢ (∀𝑧 ∈ (bday‘𝑎)∀𝑏 ∈ Ons (𝑧 = (bday‘𝑏) → 𝑏 ∈ ℕ0s) ↔ ∀𝑏 ∈ Ons ((bday‘𝑏) ∈ (bday‘𝑎) → 𝑏 ∈ ℕ0s))
63	51, 62	bitrdi 290	. . . . . . . . . . . 12 ⊢ (𝑦 = (bday‘𝑎) → (∀𝑧 ∈ 𝑦 ∀𝑏 ∈ Ons (𝑧 = (bday‘𝑏) → 𝑏 ∈ ℕ0s) ↔ ∀𝑏 ∈ Ons ((bday‘𝑏) ∈ (bday‘𝑎) → 𝑏 ∈ ℕ0s)))
64	63	imbi1d 344	. . . . . . . . . . 11 ⊢ (𝑦 = (bday‘𝑎) → ((∀𝑧 ∈ 𝑦 ∀𝑏 ∈ Ons (𝑧 = (bday‘𝑏) → 𝑏 ∈ ℕ0s) → (𝑎 ∈ Ons → 𝑎 ∈ ℕ0s)) ↔ (∀𝑏 ∈ Ons ((bday‘𝑏) ∈ (bday‘𝑎) → 𝑏 ∈ ℕ0s) → (𝑎 ∈ Ons → 𝑎 ∈ ℕ0s))))
65	50, 64	imbi12d 347	. . . . . . . . . 10 ⊢ (𝑦 = (bday‘𝑎) → ((𝑦 ∈ ω → (∀𝑧 ∈ 𝑦 ∀𝑏 ∈ Ons (𝑧 = (bday‘𝑏) → 𝑏 ∈ ℕ0s) → (𝑎 ∈ Ons → 𝑎 ∈ ℕ0s))) ↔ ((bday‘𝑎) ∈ ω → (∀𝑏 ∈ Ons ((bday‘𝑏) ∈ (bday‘𝑎) → 𝑏 ∈ ℕ0s) → (𝑎 ∈ Ons → 𝑎 ∈ ℕ0s)))))
66	49, 65	mpbiri 261	. . . . . . . . 9 ⊢ (𝑦 = (bday‘𝑎) → (𝑦 ∈ ω → (∀𝑧 ∈ 𝑦 ∀𝑏 ∈ Ons (𝑧 = (bday‘𝑏) → 𝑏 ∈ ℕ0s) → (𝑎 ∈ Ons → 𝑎 ∈ ℕ0s))))
67	66	com4l 93	. . . . . . . 8 ⊢ (𝑦 ∈ ω → (∀𝑧 ∈ 𝑦 ∀𝑏 ∈ Ons (𝑧 = (bday‘𝑏) → 𝑏 ∈ ℕ0s) → (𝑎 ∈ Ons → (𝑦 = (bday‘𝑎) → 𝑎 ∈ ℕ0s))))
68	67	ralrimdv 3164	. . . . . . 7 ⊢ (𝑦 ∈ ω → (∀𝑧 ∈ 𝑦 ∀𝑏 ∈ Ons (𝑧 = (bday‘𝑏) → 𝑏 ∈ ℕ0s) → ∀𝑎 ∈ Ons (𝑦 = (bday‘𝑎) → 𝑎 ∈ ℕ0s)))
69	10, 13, 68	omsinds 7919	. . . . . 6 ⊢ (𝑥 ∈ ω → ∀𝑎 ∈ Ons (𝑥 = (bday‘𝑎) → 𝑎 ∈ ℕ0s))
70		fveq2 6905	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → (bday‘𝑎) = (bday‘𝐴))
71	70	eqeq2d 2774	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑥 = (bday‘𝑎) ↔ 𝑥 = (bday‘𝐴)))
72		eleq1 2852	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑎 ∈ ℕ0s ↔ 𝐴 ∈ ℕ0s))
73	71, 72	imbi12d 347	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ((𝑥 = (bday‘𝑎) → 𝑎 ∈ ℕ0s) ↔ (𝑥 = (bday‘𝐴) → 𝐴 ∈ ℕ0s)))
74	73	rspccv 3577	. . . . . 6 ⊢ (∀𝑎 ∈ Ons (𝑥 = (bday‘𝑎) → 𝑎 ∈ ℕ0s) → (𝐴 ∈ Ons → (𝑥 = (bday‘𝐴) → 𝐴 ∈ ℕ0s)))
75	69, 74	syl 18	. . . . 5 ⊢ (𝑥 ∈ ω → (𝐴 ∈ Ons → (𝑥 = (bday‘𝐴) → 𝐴 ∈ ℕ0s)))
76	75	com23 87	. . . 4 ⊢ (𝑥 ∈ ω → (𝑥 = (bday‘𝐴) → (𝐴 ∈ Ons → 𝐴 ∈ ℕ0s)))
77	76	rexlimiv 3160	. . 3 ⊢ (∃𝑥 ∈ ω 𝑥 = (bday‘𝐴) → (𝐴 ∈ Ons → 𝐴 ∈ ℕ0s))
78	1, 77	sylbi 220	. 2 ⊢ ((bday‘𝐴) ∈ ω → (𝐴 ∈ Ons → 𝐴 ∈ ℕ0s))
79	78	impcom 413	1 ⊢ ((𝐴 ∈ Ons ∧ (bday‘𝐴) ∈ ω) → 𝐴 ∈ ℕ0s)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∀wal 1568   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  {crab 3416   ⊆ wss 3900  ∅c0 4281   class class class wbr 5106  Oncon0 6377  ‘cfv 6555  (class class class)co 7440  ωcom 7898  Fincfn 8998  Nocsur 28024   <s clts 28025  bdaycbday 28026   |s ccuts 28172  Ocold 28236  On_scons 28664  ℕ_0scn0s 28725

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 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772

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 4839  df-ot 4862  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-om 7899  df-1st 8022  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-1o 8501  df-2o 8502  df-nadd 8700  df-en 8999  df-dom 9000  df-fin 9002  df-no 28027  df-lts 28028  df-bday 28029  df-les 28129  df-slts 28171  df-cuts 28173  df-0s 28220  df-1s 28221  df-made 28240  df-old 28241  df-new 28242  df-left 28243  df-right 28244  df-norec 28351  df-norec2 28362  df-adds 28373  df-negs 28434  df-subs 28435  df-ons 28665  df-n0s 28727

This theorem is used by:  eln0s2  28770  onltn0s  28771