Theorem onsfi 28370

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 3216	. . 3 ⊢ (( bday ‘𝐴) ∈ ω ↔ ∃𝑥 ∈ ω 𝑥 = ( bday ‘𝐴))
2		eqeq1 2745	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → (𝑦 = ( bday ‘𝑎) ↔ 𝑧 = ( bday ‘𝑎)))
3	2	imbi1d 343	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → ((𝑦 = ( bday ‘𝑎) → 𝑎 ∈ ℕ0s) ↔ (𝑧 = ( bday ‘𝑎) → 𝑎 ∈ ℕ0s)))
4	3	ralbidv 3164	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → (∀𝑎 ∈ Ons (𝑦 = ( bday ‘𝑎) → 𝑎 ∈ ℕ0s) ↔ ∀𝑎 ∈ Ons (𝑧 = ( bday ‘𝑎) → 𝑎 ∈ ℕ0s)))
5		fveq2 6831	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑏 → ( bday ‘𝑎) = ( bday ‘𝑏))
6	5	eqeq2d 2752	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → (𝑧 = ( bday ‘𝑎) ↔ 𝑧 = ( bday ‘𝑏)))
7		eleq1 2829	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → (𝑎 ∈ ℕ0s ↔ 𝑏 ∈ ℕ0s))
8	6, 7	imbi12d 346	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → ((𝑧 = ( bday ‘𝑎) → 𝑎 ∈ ℕ0s) ↔ (𝑧 = ( bday ‘𝑏) → 𝑏 ∈ ℕ0s)))
9	8	cbvralvw 3219	. . . . . . . 8 ⊢ (∀𝑎 ∈ Ons (𝑧 = ( bday ‘𝑎) → 𝑎 ∈ ℕ0s) ↔ ∀𝑏 ∈ Ons (𝑧 = ( bday ‘𝑏) → 𝑏 ∈ ℕ0s))
10	4, 9	bitrdi 289	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (∀𝑎 ∈ Ons (𝑦 = ( bday ‘𝑎) → 𝑎 ∈ ℕ0s) ↔ ∀𝑏 ∈ Ons (𝑧 = ( bday ‘𝑏) → 𝑏 ∈ ℕ0s)))
11		eqeq1 2745	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝑦 = ( bday ‘𝑎) ↔ 𝑥 = ( bday ‘𝑎)))
12	11	imbi1d 343	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → ((𝑦 = ( bday ‘𝑎) → 𝑎 ∈ ℕ0s) ↔ (𝑥 = ( bday ‘𝑎) → 𝑎 ∈ ℕ0s)))
13	12	ralbidv 3164	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (∀𝑎 ∈ Ons (𝑦 = ( bday ‘𝑎) → 𝑎 ∈ ℕ0s) ↔ ∀𝑎 ∈ Ons (𝑥 = ( bday ‘𝑎) → 𝑎 ∈ ℕ0s)))
14		oncutlt 28278	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ Ons → 𝑎 = ({𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} |s ∅))
15	14	3ad2ant3 1142	. . . . . . . . . . . 12 ⊢ ((( bday ‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) → 𝑎 = ({𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} |s ∅))
16		onssno 28268	. . . . . . . . . . . . . . . . . . 19 ⊢ Ons ⊆ No
17		simp13 1213	. . . . . . . . . . . . . . . . . . 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 3915	. . . . . . . . . . . . . . . . . 18 ⊢ (((( bday ‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) ∧ 𝑥 ∈ Ons ∧ 𝑥 <s 𝑎) → 𝑎 ∈ No )
19		ltonold 28275	. . . . . . . . . . . . . . . . . 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 5078	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑥 → (𝑏 <s 𝑎 ↔ 𝑥 <s 𝑎))
22		simp2 1144	. . . . . . . . . . . . . . . . . 18 ⊢ (((( bday ‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) ∧ 𝑥 ∈ Ons ∧ 𝑥 <s 𝑎) → 𝑥 ∈ Ons)
23		simp3 1145	. . . . . . . . . . . . . . . . . 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 3633	. . . . . . . . . . . . . . . . 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 3918	. . . . . . . . . . . . . . . 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 27766	. . . . . . . . . . . . . . . . 17 ⊢ ( bday ‘𝑎) ∈ On
27	16, 22	sselid 3915	. . . . . . . . . . . . . . . . 17 ⊢ (((( bday ‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) ∧ 𝑥 ∈ Ons ∧ 𝑥 <s 𝑎) → 𝑥 ∈ No )
28		oldbday 27915	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑎) ∈ On ∧ 𝑥 ∈ No ) → (𝑥 ∈ ( O ‘( bday ‘𝑎)) ↔ ( bday ‘𝑥) ∈ ( bday ‘𝑎)))
29	26, 27, 28	sylancr 594	. . . . . . . . . . . . . . . 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 234	. . . . . . . . . . . . . . 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 6831	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑥 → ( bday ‘𝑏) = ( bday ‘𝑥))
32	31	eleq1d 2826	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑥 → (( bday ‘𝑏) ∈ ( bday ‘𝑎) ↔ ( bday ‘𝑥) ∈ ( bday ‘𝑎)))
33		eleq1 2829	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑥 → (𝑏 ∈ ℕ0s ↔ 𝑥 ∈ ℕ0s))
34	32, 33	imbi12d 346	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 𝑥 → ((( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s) ↔ (( bday ‘𝑥) ∈ ( bday ‘𝑎) → 𝑥 ∈ ℕ0s)))
35		simp12 1212	. . . . . . . . . . . . . . . 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 3563	. . . . . . . . . . . . . . 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 4008	. . . . . . . . . . . . 13 ⊢ ((( bday ‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) → {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} ⊆ ℕ0s)
39		oldfi 27928	. . . . . . . . . . . . . . 15 ⊢ (( bday ‘𝑎) ∈ ω → ( O ‘( bday ‘𝑎)) ∈ Fin)
40	39	3ad2ant1 1140	. . . . . . . . . . . . . 14 ⊢ ((( bday ‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) → ( O ‘( bday ‘𝑎)) ∈ Fin)
41		onno 28269	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ Ons → 𝑎 ∈ No )
42	41	3ad2ant3 1142	. . . . . . . . . . . . . . 15 ⊢ ((( bday ‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) → 𝑎 ∈ No )
43		ltonold 28275	. . . . . . . . . . . . . . 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 9173	. . . . . . . . . . . . 13 ⊢ ((( bday ‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) → {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} ∈ Fin)
46		n0fincut 28369	. . . . . . . . . . . . 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 591	. . . . . . . . . . . 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 2841	. . . . . . . . . . 11 ⊢ ((( bday ‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) → 𝑎 ∈ ℕ0s)
49	48	3exp 1126	. . . . . . . . . 10 ⊢ (( bday ‘𝑎) ∈ ω → (∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s) → (𝑎 ∈ Ons → 𝑎 ∈ ℕ0s)))
50		eleq1 2829	. . . . . . . . . . 11 ⊢ (𝑦 = ( bday ‘𝑎) → (𝑦 ∈ ω ↔ ( bday ‘𝑎) ∈ ω))
51		raleq 3296	. . . . . . . . . . . . 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 3269	. . . . . . . . . . . . . 14 ⊢ (∀𝑧 ∈ ( bday ‘𝑎)∀𝑏 ∈ Ons (𝑧 = ( bday ‘𝑏) → 𝑏 ∈ ℕ0s) ↔ ∀𝑏 ∈ Ons ∀𝑧 ∈ ( bday ‘𝑎)(𝑧 = ( bday ‘𝑏) → 𝑏 ∈ ℕ0s))
53		df-ral 3056	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑧 ∈ ( bday ‘𝑎)(𝑧 = ( bday ‘𝑏) → 𝑏 ∈ ℕ0s) ↔ ∀𝑧(𝑧 ∈ ( bday ‘𝑎) → (𝑧 = ( bday ‘𝑏) → 𝑏 ∈ ℕ0s)))
54		bi2.04 389	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ ( bday ‘𝑎) → (𝑧 = ( bday ‘𝑏) → 𝑏 ∈ ℕ0s)) ↔ (𝑧 = ( bday ‘𝑏) → (𝑧 ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s)))
55	54	albii 1827	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑧(𝑧 ∈ ( bday ‘𝑎) → (𝑧 = ( bday ‘𝑏) → 𝑏 ∈ ℕ0s)) ↔ ∀𝑧(𝑧 = ( bday ‘𝑏) → (𝑧 ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s)))
56		fvex 6844	. . . . . . . . . . . . . . . . 17 ⊢ ( bday ‘𝑏) ∈ V
57		eleq1 2829	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = ( bday ‘𝑏) → (𝑧 ∈ ( bday ‘𝑎) ↔ ( bday ‘𝑏) ∈ ( bday ‘𝑎)))
58	57	imbi1d 343	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = ( bday ‘𝑏) → ((𝑧 ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s) ↔ (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s)))
59	56, 58	ceqsalv 3472	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑧(𝑧 = ( bday ‘𝑏) → (𝑧 ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s)) ↔ (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s))
60	53, 55, 59	3bitri 299	. . . . . . . . . . . . . . 15 ⊢ (∀𝑧 ∈ ( bday ‘𝑎)(𝑧 = ( bday ‘𝑏) → 𝑏 ∈ ℕ0s) ↔ (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s))
61	60	ralbii 3087	. . . . . . . . . . . . . 14 ⊢ (∀𝑏 ∈ Ons ∀𝑧 ∈ ( bday ‘𝑎)(𝑧 = ( bday ‘𝑏) → 𝑏 ∈ ℕ0s) ↔ ∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s))
62	52, 61	bitri 277	. . . . . . . . . . . . 13 ⊢ (∀𝑧 ∈ ( bday ‘𝑎)∀𝑏 ∈ Ons (𝑧 = ( bday ‘𝑏) → 𝑏 ∈ ℕ0s) ↔ ∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s))
63	51, 62	bitrdi 289	. . . . . . . . . . . 12 ⊢ (𝑦 = ( bday ‘𝑎) → (∀𝑧 ∈ 𝑦 ∀𝑏 ∈ Ons (𝑧 = ( bday ‘𝑏) → 𝑏 ∈ ℕ0s) ↔ ∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s)))
64	63	imbi1d 343	. . . . . . . . . . 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 346	. . . . . . . . . 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 260	. . . . . . . . 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 3139	. . . . . . 7 ⊢ (𝑦 ∈ ω → (∀𝑧 ∈ 𝑦 ∀𝑏 ∈ Ons (𝑧 = ( bday ‘𝑏) → 𝑏 ∈ ℕ0s) → ∀𝑎 ∈ Ons (𝑦 = ( bday ‘𝑎) → 𝑎 ∈ ℕ0s)))
69	10, 13, 68	omsinds 7831	. . . . . 6 ⊢ (𝑥 ∈ ω → ∀𝑎 ∈ Ons (𝑥 = ( bday ‘𝑎) → 𝑎 ∈ ℕ0s))
70		fveq2 6831	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → ( bday ‘𝑎) = ( bday ‘𝐴))
71	70	eqeq2d 2752	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑥 = ( bday ‘𝑎) ↔ 𝑥 = ( bday ‘𝐴)))
72		eleq1 2829	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑎 ∈ ℕ0s ↔ 𝐴 ∈ ℕ0s))
73	71, 72	imbi12d 346	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ((𝑥 = ( bday ‘𝑎) → 𝑎 ∈ ℕ0s) ↔ (𝑥 = ( bday ‘𝐴) → 𝐴 ∈ ℕ0s)))
74	73	rspccv 3559	. . . . . 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 3135	. . 3 ⊢ (∃𝑥 ∈ ω 𝑥 = ( bday ‘𝐴) → (𝐴 ∈ Ons → 𝐴 ∈ ℕ0s))
78	1, 77	sylbi 219	. 2 ⊢ (( bday ‘𝐴) ∈ ω → (𝐴 ∈ Ons → 𝐴 ∈ ℕ0s))
79	78	impcom 409	1 ⊢ ((𝐴 ∈ Ons ∧ ( bday ‘𝐴) ∈ ω) → 𝐴 ∈ ℕ0s)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   ∧ w3a 1093  ∀wal 1546   = wceq 1548   ∈ wcel 2121  ∀wral 3055  ∃wrex 3065  {crab 3393   ⊆ wss 3885  ∅c0 4264   class class class wbr 5075  Oncon0 6314  ‘cfv 6489  (class class class)co 7360  ωcom 7810  Fincfn 8887   No csur 27625   <s clts 27626   bday cbday 27627   |s ccuts 27773   O cold 27837  On_scons 28265  ℕ_0scn0s 28326

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682  ax-ac2 10380

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-ot 4567  df-uni 4842  df-int 4881  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-se 5575  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-isom 6498  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-2o 8400  df-nadd 8596  df-er 8637  df-map 8769  df-en 8888  df-dom 8889  df-fin 8891  df-card 9858  df-acn 9861  df-ac 10033  df-no 27628  df-lts 27629  df-bday 27630  df-les 27731  df-slts 27772  df-cuts 27774  df-0s 27821  df-1s 27822  df-made 27841  df-old 27842  df-new 27843  df-left 27844  df-right 27845  df-norec 27952  df-norec2 27963  df-adds 27974  df-negs 28035  df-subs 28036  df-ons 28266  df-n0s 28328

This theorem is referenced by:  eln0s2  28371  onltn0s  28372