Theorem onsfi 28560

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 3239	. . 3 ⊢ (( bday ‘𝐴) ∈ ω ↔ ∃𝑥 ∈ ω 𝑥 = ( bday ‘𝐴))
2		eqeq1 2766	. . . . . . . . . 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 3187	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → (∀𝑎 ∈ Ons (𝑦 = ( bday ‘𝑎) → 𝑎 ∈ ℕ0s) ↔ ∀𝑎 ∈ Ons (𝑧 = ( bday ‘𝑎) → 𝑎 ∈ ℕ0s)))
5		fveq2 6881	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑏 → ( bday ‘𝑎) = ( bday ‘𝑏))
6	5	eqeq2d 2773	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → (𝑧 = ( bday ‘𝑎) ↔ 𝑧 = ( bday ‘𝑏)))
7		eleq1 2850	. . . . . . . . . 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 3242	. . . . . . . 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 2766	. . . . . . . . 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 3187	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (∀𝑎 ∈ Ons (𝑦 = ( bday ‘𝑎) → 𝑎 ∈ ℕ0s) ↔ ∀𝑎 ∈ Ons (𝑥 = ( bday ‘𝑎) → 𝑎 ∈ ℕ0s)))
14		oncutlt 28468	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ Ons → 𝑎 = ({𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} |s ∅))
15	14	3ad2ant3 1152	. . . . . . . . . . . 12 ⊢ ((( bday ‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) → 𝑎 = ({𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} |s ∅))
16		onssno 28458	. . . . . . . . . . . . . . . . . . 19 ⊢ Ons ⊆ No
17		simp13 1223	. . . . . . . . . . . . . . . . . . 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 3934	. . . . . . . . . . . . . . . . . 18 ⊢ (((( bday ‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) ∧ 𝑥 ∈ Ons ∧ 𝑥 <s 𝑎) → 𝑎 ∈ No )
19		ltonold 28465	. . . . . . . . . . . . . . . . . 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 5111	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑥 → (𝑏 <s 𝑎 ↔ 𝑥 <s 𝑎))
22		simp2 1154	. . . . . . . . . . . . . . . . . 18 ⊢ (((( bday ‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) ∧ 𝑥 ∈ Ons ∧ 𝑥 <s 𝑎) → 𝑥 ∈ Ons)
23		simp3 1155	. . . . . . . . . . . . . . . . . 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 3651	. . . . . . . . . . . . . . . . 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 3937	. . . . . . . . . . . . . . . 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 27956	. . . . . . . . . . . . . . . . 17 ⊢ ( bday ‘𝑎) ∈ On
27	16, 22	sselid 3934	. . . . . . . . . . . . . . . . 17 ⊢ (((( bday ‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) ∧ 𝑥 ∈ Ons ∧ 𝑥 <s 𝑎) → 𝑥 ∈ No )
28		oldbday 28105	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑎) ∈ On ∧ 𝑥 ∈ No ) → (𝑥 ∈ ( O ‘( bday ‘𝑎)) ↔ ( bday ‘𝑥) ∈ ( bday ‘𝑎)))
29	26, 27, 28	sylancr 598	. . . . . . . . . . . . . . . 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 6881	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑥 → ( bday ‘𝑏) = ( bday ‘𝑥))
32	31	eleq1d 2847	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑥 → (( bday ‘𝑏) ∈ ( bday ‘𝑎) ↔ ( bday ‘𝑥) ∈ ( bday ‘𝑎)))
33		eleq1 2850	. . . . . . . . . . . . . . . . 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 1222	. . . . . . . . . . . . . . . 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 4027	. . . . . . . . . . . . 13 ⊢ ((( bday ‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) → {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} ⊆ ℕ0s)
39		oldfi 28118	. . . . . . . . . . . . . . 15 ⊢ (( bday ‘𝑎) ∈ ω → ( O ‘( bday ‘𝑎)) ∈ Fin)
40	39	3ad2ant1 1150	. . . . . . . . . . . . . 14 ⊢ ((( bday ‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) → ( O ‘( bday ‘𝑎)) ∈ Fin)
41		onno 28459	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ Ons → 𝑎 ∈ No )
42	41	3ad2ant3 1152	. . . . . . . . . . . . . . 15 ⊢ ((( bday ‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) → 𝑎 ∈ No )
43		ltonold 28465	. . . . . . . . . . . . . . 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 9227	. . . . . . . . . . . . 13 ⊢ ((( bday ‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) → {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} ∈ Fin)
46		n0fincut 28559	. . . . . . . . . . . . 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 595	. . . . . . . . . . . 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 2862	. . . . . . . . . . 11 ⊢ ((( bday ‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) → 𝑎 ∈ ℕ0s)
49	48	3exp 1136	. . . . . . . . . 10 ⊢ (( bday ‘𝑎) ∈ ω → (∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s) → (𝑎 ∈ Ons → 𝑎 ∈ ℕ0s)))
50		eleq1 2850	. . . . . . . . . . 11 ⊢ (𝑦 = ( bday ‘𝑎) → (𝑦 ∈ ω ↔ ( bday ‘𝑎) ∈ ω))
51		raleq 3319	. . . . . . . . . . . . 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 3292	. . . . . . . . . . . . . 14 ⊢ (∀𝑧 ∈ ( bday ‘𝑎)∀𝑏 ∈ Ons (𝑧 = ( bday ‘𝑏) → 𝑏 ∈ ℕ0s) ↔ ∀𝑏 ∈ Ons ∀𝑧 ∈ ( bday ‘𝑎)(𝑧 = ( bday ‘𝑏) → 𝑏 ∈ ℕ0s))
53		df-ral 3079	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑧 ∈ ( bday ‘𝑎)(𝑧 = ( bday ‘𝑏) → 𝑏 ∈ ℕ0s) ↔ ∀𝑧(𝑧 ∈ ( bday ‘𝑎) → (𝑧 = ( bday ‘𝑏) → 𝑏 ∈ ℕ0s)))
54		bi2.04 391	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ ( bday ‘𝑎) → (𝑧 = ( bday ‘𝑏) → 𝑏 ∈ ℕ0s)) ↔ (𝑧 = ( bday ‘𝑏) → (𝑧 ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s)))
55	54	albii 1848	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑧(𝑧 ∈ ( bday ‘𝑎) → (𝑧 = ( bday ‘𝑏) → 𝑏 ∈ ℕ0s)) ↔ ∀𝑧(𝑧 = ( bday ‘𝑏) → (𝑧 ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s)))
56		fvex 6894	. . . . . . . . . . . . . . . . 17 ⊢ ( bday ‘𝑏) ∈ V
57		eleq1 2850	. . . . . . . . . . . . . . . . . 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 3110	. . . . . . . . . . . . . 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 3162	. . . . . . 7 ⊢ (𝑦 ∈ ω → (∀𝑧 ∈ 𝑦 ∀𝑏 ∈ Ons (𝑧 = ( bday ‘𝑏) → 𝑏 ∈ ℕ0s) → ∀𝑎 ∈ Ons (𝑦 = ( bday ‘𝑎) → 𝑎 ∈ ℕ0s)))
69	10, 13, 68	omsinds 7881	. . . . . 6 ⊢ (𝑥 ∈ ω → ∀𝑎 ∈ Ons (𝑥 = ( bday ‘𝑎) → 𝑎 ∈ ℕ0s))
70		fveq2 6881	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → ( bday ‘𝑎) = ( bday ‘𝐴))
71	70	eqeq2d 2773	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑥 = ( bday ‘𝑎) ↔ 𝑥 = ( bday ‘𝐴)))
72		eleq1 2850	. . . . . . . 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 3158	. . 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 412	1 ⊢ ((𝐴 ∈ Ons ∧ ( bday ‘𝐴) ∈ ω) → 𝐴 ∈ ℕ0s)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1102  ∀wal 1567   = wceq 1569   ∈ wcel 2142  ∀wral 3078  ∃wrex 3088  {crab 3415   ⊆ wss 3904  ∅c0 4285   class class class wbr 5108  Oncon0 6360  ‘cfv 6536  (class class class)co 7412  ωcom 7860  Fincfn 8941   No csur 27815   <s clts 27816   bday cbday 27817   |s ccuts 27963   O cold 28027  On_scons 28455  ℕ_0scn0s 28516

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734  ax-ac2 10453

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4595  df-ot 4597  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-se 5614  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  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 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7861  df-1st 7984  df-2nd 7985  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-1o 8451  df-2o 8452  df-nadd 8650  df-er 8692  df-map 8824  df-en 8942  df-dom 8943  df-fin 8945  df-card 9932  df-acn 9935  df-ac 10107  df-no 27818  df-lts 27819  df-bday 27820  df-les 27920  df-slts 27962  df-cuts 27964  df-0s 28011  df-1s 28012  df-made 28031  df-old 28032  df-new 28033  df-left 28034  df-right 28035  df-norec 28142  df-norec2 28153  df-adds 28164  df-negs 28225  df-subs 28226  df-ons 28456  df-n0s 28518

This theorem is used by:  eln0s2  28561  onltn0s  28562