Theorem onsfi 28742

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 3237	. . 3 ⊢ ((bday‘𝐴) ∈ ω ↔ ∃𝑥 ∈ ω 𝑥 = (bday‘𝐴))
2		eqeq1 2764	. . . . . . . . . 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 3185	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → (∀𝑎 ∈ Ons (𝑦 = (bday‘𝑎) → 𝑎 ∈ ℕ0s) ↔ ∀𝑎 ∈ Ons (𝑧 = (bday‘𝑎) → 𝑎 ∈ ℕ0s)))
5		fveq2 6885	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑏 → (bday‘𝑎) = (bday‘𝑏))
6	5	eqeq2d 2771	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → (𝑧 = (bday‘𝑎) ↔ 𝑧 = (bday‘𝑏)))
7		eleq1 2848	. . . . . . . . . 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 3240	. . . . . . . 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 2764	. . . . . . . . 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 3185	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (∀𝑎 ∈ Ons (𝑦 = (bday‘𝑎) → 𝑎 ∈ ℕ0s) ↔ ∀𝑎 ∈ Ons (𝑥 = (bday‘𝑎) → 𝑎 ∈ ℕ0s)))
14		oncutlt 28650	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ Ons → 𝑎 = ({𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} |s ∅))
15	14	3ad2ant3 1153	. . . . . . . . . . . 12 ⊢ (((bday‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons ((bday‘𝑏) ∈ (bday‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) → 𝑎 = ({𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} |s ∅))
16		onssno 28640	. . . . . . . . . . . . . . . . . . 19 ⊢ Ons ⊆ No
17		simp13 1224	. . . . . . . . . . . . . . . . . . 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 3906	. . . . . . . . . . . . . . . . . 18 ⊢ ((((bday‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons ((bday‘𝑏) ∈ (bday‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) ∧ 𝑥 ∈ Ons ∧ 𝑥 <s 𝑎) → 𝑎 ∈ No)
19		ltonold 28647	. . . . . . . . . . . . . . . . . 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 5105	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑥 → (𝑏 <s 𝑎 ↔ 𝑥 <s 𝑎))
22		simp2 1155	. . . . . . . . . . . . . . . . . 18 ⊢ ((((bday‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons ((bday‘𝑏) ∈ (bday‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) ∧ 𝑥 ∈ Ons ∧ 𝑥 <s 𝑎) → 𝑥 ∈ Ons)
23		simp3 1156	. . . . . . . . . . . . . . . . . 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 3646	. . . . . . . . . . . . . . . . 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 3909	. . . . . . . . . . . . . . . 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 28138	. . . . . . . . . . . . . . . . 17 ⊢ (bday‘𝑎) ∈ On
27	16, 22	sselid 3906	. . . . . . . . . . . . . . . . 17 ⊢ ((((bday‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons ((bday‘𝑏) ∈ (bday‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) ∧ 𝑥 ∈ Ons ∧ 𝑥 <s 𝑎) → 𝑥 ∈ No)
28		oldbday 28287	. . . . . . . . . . . . . . . . 17 ⊢ (((bday‘𝑎) ∈ On ∧ 𝑥 ∈ No) → (𝑥 ∈ (O‘(bday‘𝑎)) ↔ (bday‘𝑥) ∈ (bday‘𝑎)))
29	26, 27, 28	sylancr 599	. . . . . . . . . . . . . . . 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 6885	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑥 → (bday‘𝑏) = (bday‘𝑥))
32	31	eleq1d 2845	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑥 → ((bday‘𝑏) ∈ (bday‘𝑎) ↔ (bday‘𝑥) ∈ (bday‘𝑎)))
33		eleq1 2848	. . . . . . . . . . . . . . . . 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 1223	. . . . . . . . . . . . . . . 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 3577	. . . . . . . . . . . . . . 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 3999	. . . . . . . . . . . . 13 ⊢ (((bday‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons ((bday‘𝑏) ∈ (bday‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) → {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} ⊆ ℕ0s)
39		oldfi 28300	. . . . . . . . . . . . . . 15 ⊢ ((bday‘𝑎) ∈ ω → (O‘(bday‘𝑎)) ∈ Fin)
40	39	3ad2ant1 1151	. . . . . . . . . . . . . 14 ⊢ (((bday‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons ((bday‘𝑏) ∈ (bday‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) → (O‘(bday‘𝑎)) ∈ Fin)
41		onno 28641	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ Ons → 𝑎 ∈ No)
42	41	3ad2ant3 1153	. . . . . . . . . . . . . . 15 ⊢ (((bday‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons ((bday‘𝑏) ∈ (bday‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) → 𝑎 ∈ No)
43		ltonold 28647	. . . . . . . . . . . . . . 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 9260	. . . . . . . . . . . . 13 ⊢ (((bday‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons ((bday‘𝑏) ∈ (bday‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) → {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} ∈ Fin)
46		n0fincut 28741	. . . . . . . . . . . . 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 596	. . . . . . . . . . . 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 2860	. . . . . . . . . . 11 ⊢ (((bday‘𝑎) ∈ ω ∧ ∀𝑏 ∈ Ons ((bday‘𝑏) ∈ (bday‘𝑎) → 𝑏 ∈ ℕ0s) ∧ 𝑎 ∈ Ons) → 𝑎 ∈ ℕ0s)
49	48	3exp 1137	. . . . . . . . . 10 ⊢ ((bday‘𝑎) ∈ ω → (∀𝑏 ∈ Ons ((bday‘𝑏) ∈ (bday‘𝑎) → 𝑏 ∈ ℕ0s) → (𝑎 ∈ Ons → 𝑎 ∈ ℕ0s)))
50		eleq1 2848	. . . . . . . . . . 11 ⊢ (𝑦 = (bday‘𝑎) → (𝑦 ∈ ω ↔ (bday‘𝑎) ∈ ω))
51		raleq 3316	. . . . . . . . . . . . 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 3290	. . . . . . . . . . . . . 14 ⊢ (∀𝑧 ∈ (bday‘𝑎)∀𝑏 ∈ Ons (𝑧 = (bday‘𝑏) → 𝑏 ∈ ℕ0s) ↔ ∀𝑏 ∈ Ons ∀𝑧 ∈ (bday‘𝑎)(𝑧 = (bday‘𝑏) → 𝑏 ∈ ℕ0s))
53		df-ral 3077	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑧 ∈ (bday‘𝑎)(𝑧 = (bday‘𝑏) → 𝑏 ∈ ℕ0s) ↔ ∀𝑧(𝑧 ∈ (bday‘𝑎) → (𝑧 = (bday‘𝑏) → 𝑏 ∈ ℕ0s)))
54		bi2.04 392	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ (bday‘𝑎) → (𝑧 = (bday‘𝑏) → 𝑏 ∈ ℕ0s)) ↔ (𝑧 = (bday‘𝑏) → (𝑧 ∈ (bday‘𝑎) → 𝑏 ∈ ℕ0s)))
55	54	albii 1851	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑧(𝑧 ∈ (bday‘𝑎) → (𝑧 = (bday‘𝑏) → 𝑏 ∈ ℕ0s)) ↔ ∀𝑧(𝑧 = (bday‘𝑏) → (𝑧 ∈ (bday‘𝑎) → 𝑏 ∈ ℕ0s)))
56		fvex 6898	. . . . . . . . . . . . . . . . 17 ⊢ (bday‘𝑏) ∈ V
57		eleq1 2848	. . . . . . . . . . . . . . . . . 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 3489	. . . . . . . . . . . . . . . 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 3108	. . . . . . . . . . . . . 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 3160	. . . . . . 7 ⊢ (𝑦 ∈ ω → (∀𝑧 ∈ 𝑦 ∀𝑏 ∈ Ons (𝑧 = (bday‘𝑏) → 𝑏 ∈ ℕ0s) → ∀𝑎 ∈ Ons (𝑦 = (bday‘𝑎) → 𝑎 ∈ ℕ0s)))
69	10, 13, 68	omsinds 7898	. . . . . 6 ⊢ (𝑥 ∈ ω → ∀𝑎 ∈ Ons (𝑥 = (bday‘𝑎) → 𝑎 ∈ ℕ0s))
70		fveq2 6885	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → (bday‘𝑎) = (bday‘𝐴))
71	70	eqeq2d 2771	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑥 = (bday‘𝑎) ↔ 𝑥 = (bday‘𝐴)))
72		eleq1 2848	. . . . . . . 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 3573	. . . . . 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 3156	. . 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 413	1 ⊢ ((𝐴 ∈ Ons ∧ (bday‘𝐴) ∈ ω) → 𝐴 ∈ ℕ0s)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∀wal 1567   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086  {crab 3412   ⊆ wss 3895  ∅c0 4278   class class class wbr 5102  Oncon0 6362  ‘cfv 6538  (class class class)co 7420  ωcom 7877  Fincfn 8973  Nocsur 27997   <s clts 27998  bdaycbday 27999   |s ccuts 28145  Ocold 28209  On_scons 28637  ℕ_0scn0s 28698

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-ot 4592  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-2o 8477  df-nadd 8675  df-en 8974  df-dom 8975  df-fin 8977  df-no 28000  df-lts 28001  df-bday 28002  df-les 28102  df-slts 28144  df-cuts 28146  df-0s 28193  df-1s 28194  df-made 28213  df-old 28214  df-new 28215  df-left 28216  df-right 28217  df-norec 28324  df-norec2 28335  df-adds 28346  df-negs 28407  df-subs 28408  df-ons 28638  df-n0s 28700

This theorem is used by:  eln0s2  28743  onltn0s  28744