Theorem onsfi 28366

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 3213	. . 3 ⊢ (( bday ‘𝐴) ∈ ω ↔ ∃𝑥 ∈ ω 𝑥 = ( bday ‘𝐴))
2		eqeq1 2741	. . . . . . . . . 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 3161	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → (∀𝑎 ∈ Ons (𝑦 = ( bday ‘𝑎) → 𝑎 ∈ ℕ0s) ↔ ∀𝑎 ∈ Ons (𝑧 = ( bday ‘𝑎) → 𝑎 ∈ ℕ0s)))
5		fveq2 6836	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑏 → ( bday ‘𝑎) = ( bday ‘𝑏))
6	5	eqeq2d 2748	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → (𝑧 = ( bday ‘𝑎) ↔ 𝑧 = ( bday ‘𝑏)))
7		eleq1 2825	. . . . . . . . . 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 3216	. . . . . . . 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 2741	. . . . . . . . 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 3161	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (∀𝑎 ∈ Ons (𝑦 = ( bday ‘𝑎) → 𝑎 ∈ ℕ0s) ↔ ∀𝑎 ∈ Ons (𝑥 = ( bday ‘𝑎) → 𝑎 ∈ ℕ0s)))
14		oncutlt 28274	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ Ons → 𝑎 = ({𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} |s ∅))
15	14	3ad2ant3 1136	. . . . . . . . . . . 12 ⊢ ((( bday ‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) → 𝑎 = ({𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} |s ∅))
16		onssno 28264	. . . . . . . . . . . . . . . . . . 19 ⊢ Ons ⊆ No
17		simp13 1207	. . . . . . . . . . . . . . . . . . 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 3920	. . . . . . . . . . . . . . . . . 18 ⊢ (((( bday ‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) ∧ 𝑥 ∈ Ons ∧ 𝑥 <s 𝑎) → 𝑎 ∈ No )
19		ltonold 28271	. . . . . . . . . . . . . . . . . 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 5089	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑥 → (𝑏 <s 𝑎 ↔ 𝑥 <s 𝑎))
22		simp2 1138	. . . . . . . . . . . . . . . . . 18 ⊢ (((( bday ‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) ∧ 𝑥 ∈ Ons ∧ 𝑥 <s 𝑎) → 𝑥 ∈ Ons)
23		simp3 1139	. . . . . . . . . . . . . . . . . 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 3637	. . . . . . . . . . . . . . . . 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 3923	. . . . . . . . . . . . . . . 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 27762	. . . . . . . . . . . . . . . . 17 ⊢ ( bday ‘𝑎) ∈ On
27	16, 22	sselid 3920	. . . . . . . . . . . . . . . . 17 ⊢ (((( bday ‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) ∧ 𝑥 ∈ Ons ∧ 𝑥 <s 𝑎) → 𝑥 ∈ No )
28		oldbday 27911	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑎) ∈ On ∧ 𝑥 ∈ No ) → (𝑥 ∈ ( O ‘( bday ‘𝑎)) ↔ ( bday ‘𝑥) ∈ ( bday ‘𝑎)))
29	26, 27, 28	sylancr 588	. . . . . . . . . . . . . . . 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 6836	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑥 → ( bday ‘𝑏) = ( bday ‘𝑥))
32	31	eleq1d 2822	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑥 → (( bday ‘𝑏) ∈ ( bday ‘𝑎) ↔ ( bday ‘𝑥) ∈ ( bday ‘𝑎)))
33		eleq1 2825	. . . . . . . . . . . . . . . . 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 1206	. . . . . . . . . . . . . . . 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 3566	. . . . . . . . . . . . . . 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 4015	. . . . . . . . . . . . 13 ⊢ ((( bday ‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) → {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} ⊆ ℕ0s)
39		oldfi 27924	. . . . . . . . . . . . . . 15 ⊢ (( bday ‘𝑎) ∈ ω → ( O ‘( bday ‘𝑎)) ∈ Fin)
40	39	3ad2ant1 1134	. . . . . . . . . . . . . 14 ⊢ ((( bday ‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) → ( O ‘( bday ‘𝑎)) ∈ Fin)
41		onno 28265	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ Ons → 𝑎 ∈ No )
42	41	3ad2ant3 1136	. . . . . . . . . . . . . . 15 ⊢ ((( bday ‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) → 𝑎 ∈ No )
43		ltonold 28271	. . . . . . . . . . . . . . 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 9174	. . . . . . . . . . . . 13 ⊢ ((( bday ‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) → {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} ∈ Fin)
46		n0fincut 28365	. . . . . . . . . . . . 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 585	. . . . . . . . . . . 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 2837	. . . . . . . . . . 11 ⊢ ((( bday ‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) → 𝑎 ∈ ℕ0s)
49	48	3exp 1120	. . . . . . . . . 10 ⊢ (( bday ‘𝑎) ∈ ω → (∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s) → (𝑎 ∈ Ons → 𝑎 ∈ ℕ0s)))
50		eleq1 2825	. . . . . . . . . . 11 ⊢ (𝑦 = ( bday ‘𝑎) → (𝑦 ∈ ω ↔ ( bday ‘𝑎) ∈ ω))
51		raleq 3293	. . . . . . . . . . . . 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 3266	. . . . . . . . . . . . . 14 ⊢ (∀𝑧 ∈ ( bday ‘𝑎)∀𝑏 ∈ Ons (𝑧 = ( bday ‘𝑏) → 𝑏 ∈ ℕ0s) ↔ ∀𝑏 ∈ Ons ∀𝑧 ∈ ( bday ‘𝑎)(𝑧 = ( bday ‘𝑏) → 𝑏 ∈ ℕ0s))
53		df-ral 3053	. . . . . . . . . . . . . . . 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 1821	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑧(𝑧 ∈ ( bday ‘𝑎) → (𝑧 = ( bday ‘𝑏) → 𝑏 ∈ ℕ0s)) ↔ ∀𝑧(𝑧 = ( bday ‘𝑏) → (𝑧 ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s)))
56		fvex 6849	. . . . . . . . . . . . . . . . 17 ⊢ ( bday ‘𝑏) ∈ V
57		eleq1 2825	. . . . . . . . . . . . . . . . . 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 3470	. . . . . . . . . . . . . . . 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 3084	. . . . . . . . . . . . . 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 3136	. . . . . . 7 ⊢ (𝑦 ∈ ω → (∀𝑧 ∈ 𝑦 ∀𝑏 ∈ Ons (𝑧 = ( bday ‘𝑏) → 𝑏 ∈ ℕ0s) → ∀𝑎 ∈ Ons (𝑦 = ( bday ‘𝑎) → 𝑎 ∈ ℕ0s)))
69	10, 13, 68	omsinds 7833	. . . . . 6 ⊢ (𝑥 ∈ ω → ∀𝑎 ∈ Ons (𝑥 = ( bday ‘𝑎) → 𝑎 ∈ ℕ0s))
70		fveq2 6836	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → ( bday ‘𝑎) = ( bday ‘𝐴))
71	70	eqeq2d 2748	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑥 = ( bday ‘𝑎) ↔ 𝑥 = ( bday ‘𝐴)))
72		eleq1 2825	. . . . . . . 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 3562	. . . . . 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 3132	. . 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 1087  ∀wal 1540   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  {crab 3390   ⊆ wss 3890  ∅c0 4274   class class class wbr 5086  Oncon0 6319  ‘cfv 6494  (class class class)co 7362  ωcom 7812  Fincfn 8888   No csur 27621   <s clts 27622   bday cbday 27623   |s ccuts 27769   O cold 27833  On_scons 28261  ℕ_0scn0s 28322

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 5213  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684  ax-ac2 10380

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 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-ot 4577  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-se 5580  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-pred 6261  df-ord 6322  df-on 6323  df-lim 6324  df-suc 6325  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-isom 6503  df-riota 7319  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7813  df-1st 7937  df-2nd 7938  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344  df-1o 8400  df-2o 8401  df-nadd 8597  df-er 8638  df-map 8770  df-en 8889  df-dom 8890  df-fin 8892  df-card 9858  df-acn 9861  df-ac 10033  df-no 27624  df-lts 27625  df-bday 27626  df-les 27727  df-slts 27768  df-cuts 27770  df-0s 27817  df-1s 27818  df-made 27837  df-old 27838  df-new 27839  df-left 27840  df-right 27841  df-norec 27948  df-norec2 27959  df-adds 27970  df-negs 28031  df-subs 28032  df-ons 28262  df-n0s 28324

This theorem is referenced by:  eln0s2  28367  onltn0s  28368