Theorem onsfi 28254

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 3204	. . 3 ⊢ (( bday ‘𝐴) ∈ ω ↔ ∃𝑥 ∈ ω 𝑥 = ( bday ‘𝐴))
2		eqeq1 2733	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → (𝑦 = ( bday ‘𝑎) ↔ 𝑧 = ( bday ‘𝑎)))
3	2	imbi1d 341	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → ((𝑦 = ( bday ‘𝑎) → 𝑎 ∈ ℕ0s) ↔ (𝑧 = ( bday ‘𝑎) → 𝑎 ∈ ℕ0s)))
4	3	ralbidv 3152	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → (∀𝑎 ∈ Ons (𝑦 = ( bday ‘𝑎) → 𝑎 ∈ ℕ0s) ↔ ∀𝑎 ∈ Ons (𝑧 = ( bday ‘𝑎) → 𝑎 ∈ ℕ0s)))
5		fveq2 6822	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑏 → ( bday ‘𝑎) = ( bday ‘𝑏))
6	5	eqeq2d 2740	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → (𝑧 = ( bday ‘𝑎) ↔ 𝑧 = ( bday ‘𝑏)))
7		eleq1 2816	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → (𝑎 ∈ ℕ0s ↔ 𝑏 ∈ ℕ0s))
8	6, 7	imbi12d 344	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → ((𝑧 = ( bday ‘𝑎) → 𝑎 ∈ ℕ0s) ↔ (𝑧 = ( bday ‘𝑏) → 𝑏 ∈ ℕ0s)))
9	8	cbvralvw 3207	. . . . . . . 8 ⊢ (∀𝑎 ∈ Ons (𝑧 = ( bday ‘𝑎) → 𝑎 ∈ ℕ0s) ↔ ∀𝑏 ∈ Ons (𝑧 = ( bday ‘𝑏) → 𝑏 ∈ ℕ0s))
10	4, 9	bitrdi 287	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (∀𝑎 ∈ Ons (𝑦 = ( bday ‘𝑎) → 𝑎 ∈ ℕ0s) ↔ ∀𝑏 ∈ Ons (𝑧 = ( bday ‘𝑏) → 𝑏 ∈ ℕ0s)))
11		eqeq1 2733	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝑦 = ( bday ‘𝑎) ↔ 𝑥 = ( bday ‘𝑎)))
12	11	imbi1d 341	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → ((𝑦 = ( bday ‘𝑎) → 𝑎 ∈ ℕ0s) ↔ (𝑥 = ( bday ‘𝑎) → 𝑎 ∈ ℕ0s)))
13	12	ralbidv 3152	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (∀𝑎 ∈ Ons (𝑦 = ( bday ‘𝑎) → 𝑎 ∈ ℕ0s) ↔ ∀𝑎 ∈ Ons (𝑥 = ( bday ‘𝑎) → 𝑎 ∈ ℕ0s)))
14		onscutlt 28172	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ Ons → 𝑎 = ({𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} |s ∅))
15	14	3ad2ant3 1135	. . . . . . . . . . . 12 ⊢ ((( bday ‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) → 𝑎 = ({𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} |s ∅))
16		onssno 28162	. . . . . . . . . . . . . . . . . . 19 ⊢ Ons ⊆ No
17		simp13 1206	. . . . . . . . . . . . . . . . . . 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 3933	. . . . . . . . . . . . . . . . . 18 ⊢ (((( bday ‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) ∧ 𝑥 ∈ Ons ∧ 𝑥 <s 𝑎) → 𝑎 ∈ No )
19		sltonold 28169	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 ∈ No → {𝑏 ∈ Ons ∣ 𝑏 <s 𝑎} ⊆ ( O ‘( bday ‘𝑎)))
20	18, 19	syl 17	. . . . . . . . . . . . . . . . 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 5095	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑥 → (𝑏 <s 𝑎 ↔ 𝑥 <s 𝑎))
22		simp2 1137	. . . . . . . . . . . . . . . . . 18 ⊢ (((( bday ‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) ∧ 𝑥 ∈ Ons ∧ 𝑥 <s 𝑎) → 𝑥 ∈ Ons)
23		simp3 1138	. . . . . . . . . . . . . . . . . 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 3936	. . . . . . . . . . . . . . . 16 ⊢ (((( bday ‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) ∧ 𝑥 ∈ Ons ∧ 𝑥 <s 𝑎) → 𝑥 ∈ ( O ‘( bday ‘𝑎)))
26		bdayelon 27686	. . . . . . . . . . . . . . . . 17 ⊢ ( bday ‘𝑎) ∈ On
27	16, 22	sselid 3933	. . . . . . . . . . . . . . . . 17 ⊢ (((( bday ‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) ∧ 𝑥 ∈ Ons ∧ 𝑥 <s 𝑎) → 𝑥 ∈ No )
28		oldbday 27817	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑎) ∈ On ∧ 𝑥 ∈ No ) → (𝑥 ∈ ( O ‘( bday ‘𝑎)) ↔ ( bday ‘𝑥) ∈ ( bday ‘𝑎)))
29	26, 27, 28	sylancr 587	. . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . 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 6822	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑥 → ( bday ‘𝑏) = ( bday ‘𝑥))
32	31	eleq1d 2813	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑥 → (( bday ‘𝑏) ∈ ( bday ‘𝑎) ↔ ( bday ‘𝑥) ∈ ( bday ‘𝑎)))
33		eleq1 2816	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑥 → (𝑏 ∈ ℕ0s ↔ 𝑥 ∈ ℕ0s))
34	32, 33	imbi12d 344	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 𝑥 → ((( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s) ↔ (( bday ‘𝑥) ∈ ( bday ‘𝑎) → 𝑥 ∈ ℕ0s)))
35		simp12 1205	. . . . . . . . . . . . . . . 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 3578	. . . . . . . . . . . . . . 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 15	. . . . . . . . . . . . . 14 ⊢ (((( bday ‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) ∧ 𝑥 ∈ Ons ∧ 𝑥 <s 𝑎) → 𝑥 ∈ ℕ0s)
38	37	rabssdv 4026	. . . . . . . . . . . . 13 ⊢ ((( bday ‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) → {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} ⊆ ℕ0s)
39		oldfi 27830	. . . . . . . . . . . . . . 15 ⊢ (( bday ‘𝑎) ∈ ω → ( O ‘( bday ‘𝑎)) ∈ Fin)
40	39	3ad2ant1 1133	. . . . . . . . . . . . . 14 ⊢ ((( bday ‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) → ( O ‘( bday ‘𝑎)) ∈ Fin)
41		onsno 28163	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ Ons → 𝑎 ∈ No )
42	41	3ad2ant3 1135	. . . . . . . . . . . . . . 15 ⊢ ((( bday ‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) → 𝑎 ∈ No )
43		sltonold 28169	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ No → {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} ⊆ ( O ‘( bday ‘𝑎)))
44	42, 43	syl 17	. . . . . . . . . . . . . 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 9158	. . . . . . . . . . . . 13 ⊢ ((( bday ‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) → {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} ∈ Fin)
46		n0sfincut 28253	. . . . . . . . . . . . 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 584	. . . . . . . . . . . 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 2828	. . . . . . . . . . 11 ⊢ ((( bday ‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) → 𝑎 ∈ ℕ0s)
49	48	3exp 1119	. . . . . . . . . 10 ⊢ (( bday ‘𝑎) ∈ ω → (∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s) → (𝑎 ∈ Ons → 𝑎 ∈ ℕ0s)))
50		eleq1 2816	. . . . . . . . . . 11 ⊢ (𝑦 = ( bday ‘𝑎) → (𝑦 ∈ ω ↔ ( bday ‘𝑎) ∈ ω))
51		raleq 3286	. . . . . . . . . . . . 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 3257	. . . . . . . . . . . . . 14 ⊢ (∀𝑧 ∈ ( bday ‘𝑎)∀𝑏 ∈ Ons (𝑧 = ( bday ‘𝑏) → 𝑏 ∈ ℕ0s) ↔ ∀𝑏 ∈ Ons ∀𝑧 ∈ ( bday ‘𝑎)(𝑧 = ( bday ‘𝑏) → 𝑏 ∈ ℕ0s))
53		df-ral 3045	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑧 ∈ ( bday ‘𝑎)(𝑧 = ( bday ‘𝑏) → 𝑏 ∈ ℕ0s) ↔ ∀𝑧(𝑧 ∈ ( bday ‘𝑎) → (𝑧 = ( bday ‘𝑏) → 𝑏 ∈ ℕ0s)))
54		bi2.04 387	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ ( bday ‘𝑎) → (𝑧 = ( bday ‘𝑏) → 𝑏 ∈ ℕ0s)) ↔ (𝑧 = ( bday ‘𝑏) → (𝑧 ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s)))
55	54	albii 1819	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑧(𝑧 ∈ ( bday ‘𝑎) → (𝑧 = ( bday ‘𝑏) → 𝑏 ∈ ℕ0s)) ↔ ∀𝑧(𝑧 = ( bday ‘𝑏) → (𝑧 ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s)))
56		fvex 6835	. . . . . . . . . . . . . . . . 17 ⊢ ( bday ‘𝑏) ∈ V
57		eleq1 2816	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = ( bday ‘𝑏) → (𝑧 ∈ ( bday ‘𝑎) ↔ ( bday ‘𝑏) ∈ ( bday ‘𝑎)))
58	57	imbi1d 341	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = ( bday ‘𝑏) → ((𝑧 ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s) ↔ (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s)))
59	56, 58	ceqsalv 3476	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑧(𝑧 = ( bday ‘𝑏) → (𝑧 ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s)) ↔ (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s))
60	53, 55, 59	3bitri 297	. . . . . . . . . . . . . . 15 ⊢ (∀𝑧 ∈ ( bday ‘𝑎)(𝑧 = ( bday ‘𝑏) → 𝑏 ∈ ℕ0s) ↔ (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s))
61	60	ralbii 3075	. . . . . . . . . . . . . 14 ⊢ (∀𝑏 ∈ Ons ∀𝑧 ∈ ( bday ‘𝑎)(𝑧 = ( bday ‘𝑏) → 𝑏 ∈ ℕ0s) ↔ ∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s))
62	52, 61	bitri 275	. . . . . . . . . . . . 13 ⊢ (∀𝑧 ∈ ( bday ‘𝑎)∀𝑏 ∈ Ons (𝑧 = ( bday ‘𝑏) → 𝑏 ∈ ℕ0s) ↔ ∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s))
63	51, 62	bitrdi 287	. . . . . . . . . . . 12 ⊢ (𝑦 = ( bday ‘𝑎) → (∀𝑧 ∈ 𝑦 ∀𝑏 ∈ Ons (𝑧 = ( bday ‘𝑏) → 𝑏 ∈ ℕ0s) ↔ ∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s)))
64	63	imbi1d 341	. . . . . . . . . . 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 344	. . . . . . . . . 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 258	. . . . . . . . 9 ⊢ (𝑦 = ( bday ‘𝑎) → (𝑦 ∈ ω → (∀𝑧 ∈ 𝑦 ∀𝑏 ∈ Ons (𝑧 = ( bday ‘𝑏) → 𝑏 ∈ ℕ0s) → (𝑎 ∈ Ons → 𝑎 ∈ ℕ0s))))
67	66	com4l 92	. . . . . . . 8 ⊢ (𝑦 ∈ ω → (∀𝑧 ∈ 𝑦 ∀𝑏 ∈ Ons (𝑧 = ( bday ‘𝑏) → 𝑏 ∈ ℕ0s) → (𝑎 ∈ Ons → (𝑦 = ( bday ‘𝑎) → 𝑎 ∈ ℕ0s))))
68	67	ralrimdv 3127	. . . . . . 7 ⊢ (𝑦 ∈ ω → (∀𝑧 ∈ 𝑦 ∀𝑏 ∈ Ons (𝑧 = ( bday ‘𝑏) → 𝑏 ∈ ℕ0s) → ∀𝑎 ∈ Ons (𝑦 = ( bday ‘𝑎) → 𝑎 ∈ ℕ0s)))
69	10, 13, 68	omsinds 7820	. . . . . 6 ⊢ (𝑥 ∈ ω → ∀𝑎 ∈ Ons (𝑥 = ( bday ‘𝑎) → 𝑎 ∈ ℕ0s))
70		fveq2 6822	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → ( bday ‘𝑎) = ( bday ‘𝐴))
71	70	eqeq2d 2740	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑥 = ( bday ‘𝑎) ↔ 𝑥 = ( bday ‘𝐴)))
72		eleq1 2816	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑎 ∈ ℕ0s ↔ 𝐴 ∈ ℕ0s))
73	71, 72	imbi12d 344	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ((𝑥 = ( bday ‘𝑎) → 𝑎 ∈ ℕ0s) ↔ (𝑥 = ( bday ‘𝐴) → 𝐴 ∈ ℕ0s)))
74	73	rspccv 3574	. . . . . 6 ⊢ (∀𝑎 ∈ Ons (𝑥 = ( bday ‘𝑎) → 𝑎 ∈ ℕ0s) → (𝐴 ∈ Ons → (𝑥 = ( bday ‘𝐴) → 𝐴 ∈ ℕ0s)))
75	69, 74	syl 17	. . . . 5 ⊢ (𝑥 ∈ ω → (𝐴 ∈ Ons → (𝑥 = ( bday ‘𝐴) → 𝐴 ∈ ℕ0s)))
76	75	com23 86	. . . 4 ⊢ (𝑥 ∈ ω → (𝑥 = ( bday ‘𝐴) → (𝐴 ∈ Ons → 𝐴 ∈ ℕ0s)))
77	76	rexlimiv 3123	. . 3 ⊢ (∃𝑥 ∈ ω 𝑥 = ( bday ‘𝐴) → (𝐴 ∈ Ons → 𝐴 ∈ ℕ0s))
78	1, 77	sylbi 217	. 2 ⊢ (( bday ‘𝐴) ∈ ω → (𝐴 ∈ Ons → 𝐴 ∈ ℕ0s))
79	78	impcom 407	1 ⊢ ((𝐴 ∈ Ons ∧ ( bday ‘𝐴) ∈ ω) → 𝐴 ∈ ℕ0s)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086  ∀wal 1538   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  {crab 3394   ⊆ wss 3903  ∅c0 4284   class class class wbr 5092  Oncon0 6307  ‘cfv 6482  (class class class)co 7349  ωcom 7799  Fincfn 8872   No csur 27549   <s cslt 27550   bday cbday 27551   |s cscut 27693   O cold 27755  On_scons 28159  ℕ_0scnn0s 28213

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-ac2 10357

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-ot 4586  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-isom 6491  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-nadd 8584  df-er 8625  df-map 8755  df-en 8873  df-dom 8874  df-fin 8876  df-card 9835  df-acn 9838  df-ac 10010  df-no 27552  df-slt 27553  df-bday 27554  df-sle 27655  df-sslt 27692  df-scut 27694  df-0s 27739  df-1s 27740  df-made 27759  df-old 27760  df-new 27761  df-left 27762  df-right 27763  df-norec 27852  df-norec2 27863  df-adds 27874  df-negs 27934  df-subs 27935  df-ons 28160  df-n0s 28215

This theorem is referenced by:  onltn0s  28255