Theorem onsfi 28515

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 3246	. . 3 ⊢ (( bday ‘𝐴) ∈ ω ↔ ∃𝑥 ∈ ω 𝑥 = ( bday ‘𝐴))
2		eqeq1 2773	. . . . . . . . . 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 3194	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → (∀𝑎 ∈ Ons (𝑦 = ( bday ‘𝑎) → 𝑎 ∈ ℕ0s) ↔ ∀𝑎 ∈ Ons (𝑧 = ( bday ‘𝑎) → 𝑎 ∈ ℕ0s)))
5		fveq2 6882	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑏 → ( bday ‘𝑎) = ( bday ‘𝑏))
6	5	eqeq2d 2780	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → (𝑧 = ( bday ‘𝑎) ↔ 𝑧 = ( bday ‘𝑏)))
7		eleq1 2857	. . . . . . . . . 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 3249	. . . . . . . 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 2773	. . . . . . . . 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 3194	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (∀𝑎 ∈ Ons (𝑦 = ( bday ‘𝑎) → 𝑎 ∈ ℕ0s) ↔ ∀𝑎 ∈ Ons (𝑥 = ( bday ‘𝑎) → 𝑎 ∈ ℕ0s)))
14		oncutlt 28423	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ Ons → 𝑎 = ({𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} |s ∅))
15	14	3ad2ant3 1151	. . . . . . . . . . . 12 ⊢ ((( bday ‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) → 𝑎 = ({𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} |s ∅))
16		onssno 28413	. . . . . . . . . . . . . . . . . . 19 ⊢ Ons ⊆ No
17		simp13 1222	. . . . . . . . . . . . . . . . . . 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 3941	. . . . . . . . . . . . . . . . . 18 ⊢ (((( bday ‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) ∧ 𝑥 ∈ Ons ∧ 𝑥 <s 𝑎) → 𝑎 ∈ No )
19		ltonold 28420	. . . . . . . . . . . . . . . . . 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 5114	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑥 → (𝑏 <s 𝑎 ↔ 𝑥 <s 𝑎))
22		simp2 1153	. . . . . . . . . . . . . . . . . 18 ⊢ (((( bday ‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) ∧ 𝑥 ∈ Ons ∧ 𝑥 <s 𝑎) → 𝑥 ∈ Ons)
23		simp3 1154	. . . . . . . . . . . . . . . . . 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 3659	. . . . . . . . . . . . . . . . 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 3944	. . . . . . . . . . . . . . . 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 27911	. . . . . . . . . . . . . . . . 17 ⊢ ( bday ‘𝑎) ∈ On
27	16, 22	sselid 3941	. . . . . . . . . . . . . . . . 17 ⊢ (((( bday ‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) ∧ 𝑥 ∈ Ons ∧ 𝑥 <s 𝑎) → 𝑥 ∈ No )
28		oldbday 28060	. . . . . . . . . . . . . . . . 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 6882	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑥 → ( bday ‘𝑏) = ( bday ‘𝑥))
32	31	eleq1d 2854	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑥 → (( bday ‘𝑏) ∈ ( bday ‘𝑎) ↔ ( bday ‘𝑥) ∈ ( bday ‘𝑎)))
33		eleq1 2857	. . . . . . . . . . . . . . . . 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 1221	. . . . . . . . . . . . . . . 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 3589	. . . . . . . . . . . . . . 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 4034	. . . . . . . . . . . . 13 ⊢ ((( bday ‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) → {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} ⊆ ℕ0s)
39		oldfi 28073	. . . . . . . . . . . . . . 15 ⊢ (( bday ‘𝑎) ∈ ω → ( O ‘( bday ‘𝑎)) ∈ Fin)
40	39	3ad2ant1 1149	. . . . . . . . . . . . . 14 ⊢ ((( bday ‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) → ( O ‘( bday ‘𝑎)) ∈ Fin)
41		onno 28414	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ Ons → 𝑎 ∈ No )
42	41	3ad2ant3 1151	. . . . . . . . . . . . . . 15 ⊢ ((( bday ‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) → 𝑎 ∈ No )
43		ltonold 28420	. . . . . . . . . . . . . . 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 9229	. . . . . . . . . . . . 13 ⊢ ((( bday ‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) → {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} ∈ Fin)
46		n0fincut 28514	. . . . . . . . . . . . 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 2869	. . . . . . . . . . 11 ⊢ ((( bday ‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) → 𝑎 ∈ ℕ0s)
49	48	3exp 1135	. . . . . . . . . 10 ⊢ (( bday ‘𝑎) ∈ ω → (∀𝑏 ∈ Ons (( bday ‘𝑏) ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s) → (𝑎 ∈ Ons → 𝑎 ∈ ℕ0s)))
50		eleq1 2857	. . . . . . . . . . 11 ⊢ (𝑦 = ( bday ‘𝑎) → (𝑦 ∈ ω ↔ ( bday ‘𝑎) ∈ ω))
51		raleq 3326	. . . . . . . . . . . . 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 3299	. . . . . . . . . . . . . 14 ⊢ (∀𝑧 ∈ ( bday ‘𝑎)∀𝑏 ∈ Ons (𝑧 = ( bday ‘𝑏) → 𝑏 ∈ ℕ0s) ↔ ∀𝑏 ∈ Ons ∀𝑧 ∈ ( bday ‘𝑎)(𝑧 = ( bday ‘𝑏) → 𝑏 ∈ ℕ0s))
53		df-ral 3086	. . . . . . . . . . . . . . . 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 1846	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑧(𝑧 ∈ ( bday ‘𝑎) → (𝑧 = ( bday ‘𝑏) → 𝑏 ∈ ℕ0s)) ↔ ∀𝑧(𝑧 = ( bday ‘𝑏) → (𝑧 ∈ ( bday ‘𝑎) → 𝑏 ∈ ℕ0s)))
56		fvex 6895	. . . . . . . . . . . . . . . . 17 ⊢ ( bday ‘𝑏) ∈ V
57		eleq1 2857	. . . . . . . . . . . . . . . . . 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 3500	. . . . . . . . . . . . . . . 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 3117	. . . . . . . . . . . . . 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 3169	. . . . . . 7 ⊢ (𝑦 ∈ ω → (∀𝑧 ∈ 𝑦 ∀𝑏 ∈ Ons (𝑧 = ( bday ‘𝑏) → 𝑏 ∈ ℕ0s) → ∀𝑎 ∈ Ons (𝑦 = ( bday ‘𝑎) → 𝑎 ∈ ℕ0s)))
69	10, 13, 68	omsinds 7883	. . . . . 6 ⊢ (𝑥 ∈ ω → ∀𝑎 ∈ Ons (𝑥 = ( bday ‘𝑎) → 𝑎 ∈ ℕ0s))
70		fveq2 6882	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → ( bday ‘𝑎) = ( bday ‘𝐴))
71	70	eqeq2d 2780	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑥 = ( bday ‘𝑎) ↔ 𝑥 = ( bday ‘𝐴)))
72		eleq1 2857	. . . . . . . 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 3585	. . . . . 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 3165	. . 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)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101  ∀wal 1565   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095  {crab 3422   ⊆ wss 3911  ∅c0 4292   class class class wbr 5111  Oncon0 6361  ‘cfv 6537  (class class class)co 7411  ωcom 7862  Fincfn 8943   No csur 27770   <s clts 27771   bday cbday 27772   |s ccuts 27918   O cold 27982  On_scons 28410  ℕ_0scn0s 28471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-ac2 10447

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3375  df-reu 3376  df-rab 3423  df-v 3463  df-sbc 3752  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-ot 4601  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-se 5616  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-2o 8454  df-nadd 8652  df-er 8694  df-map 8826  df-en 8944  df-dom 8945  df-fin 8947  df-card 9925  df-acn 9928  df-ac 10100  df-no 27773  df-lts 27774  df-bday 27775  df-les 27875  df-slts 27917  df-cuts 27919  df-0s 27966  df-1s 27967  df-made 27986  df-old 27987  df-new 27988  df-left 27989  df-right 27990  df-norec 28097  df-norec2 28108  df-adds 28119  df-negs 28180  df-subs 28181  df-ons 28411  df-n0s 28473

This theorem is referenced by:  eln0s2  28516  onltn0s  28517