Theorem bday1s 34025

Description: The birthday of surreal one is ordinal one. (Contributed by Scott Fenton, 8-Aug-2024.)

Assertion

Ref	Expression
bday1s	⊢ ( bday ‘ 1s ) = 1o

Proof of Theorem bday1s

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-1s 34019	. . 3 ⊢ 1s = ({ 0s } |s ∅)
2	1	fveq2i 6777	. 2 ⊢ ( bday ‘ 1s ) = ( bday ‘({ 0s } |s ∅))
3		0sno 34020	. . . . . . 7 ⊢ 0s ∈ No
4		snelpwi 5360	. . . . . . 7 ⊢ ( 0s ∈ No → { 0s } ∈ 𝒫 No )
5	3, 4	ax-mp 5	. . . . . 6 ⊢ { 0s } ∈ 𝒫 No
6		nulssgt 33992	. . . . . 6 ⊢ ({ 0s } ∈ 𝒫 No → { 0s } <<s ∅)
7	5, 6	ax-mp 5	. . . . 5 ⊢ { 0s } <<s ∅
8		scutbdaybnd2 34010	. . . . 5 ⊢ ({ 0s } <<s ∅ → ( bday ‘({ 0s } |s ∅)) ⊆ suc ∪ ( bday “ ({ 0s } ∪ ∅)))
9	7, 8	ax-mp 5	. . . 4 ⊢ ( bday ‘({ 0s } |s ∅)) ⊆ suc ∪ ( bday “ ({ 0s } ∪ ∅))
10		un0 4324	. . . . . . . . . 10 ⊢ ({ 0s } ∪ ∅) = { 0s }
11	10	imaeq2i 5967	. . . . . . . . 9 ⊢ ( bday “ ({ 0s } ∪ ∅)) = ( bday “ { 0s })
12		bdayfn 33968	. . . . . . . . . 10 ⊢ bday Fn No
13		fnsnfv 6847	. . . . . . . . . 10 ⊢ (( bday Fn No ∧ 0s ∈ No ) → {( bday ‘ 0s )} = ( bday “ { 0s }))
14	12, 3, 13	mp2an 689	. . . . . . . . 9 ⊢ {( bday ‘ 0s )} = ( bday “ { 0s })
15		bday0s 34022	. . . . . . . . . 10 ⊢ ( bday ‘ 0s ) = ∅
16	15	sneqi 4572	. . . . . . . . 9 ⊢ {( bday ‘ 0s )} = {∅}
17	11, 14, 16	3eqtr2i 2772	. . . . . . . 8 ⊢ ( bday “ ({ 0s } ∪ ∅)) = {∅}
18	17	unieqi 4852	. . . . . . 7 ⊢ ∪ ( bday “ ({ 0s } ∪ ∅)) = ∪ {∅}
19		0ex 5231	. . . . . . . 8 ⊢ ∅ ∈ V
20	19	unisn 4861	. . . . . . 7 ⊢ ∪ {∅} = ∅
21	18, 20	eqtri 2766	. . . . . 6 ⊢ ∪ ( bday “ ({ 0s } ∪ ∅)) = ∅
22		suceq 6331	. . . . . 6 ⊢ (∪ ( bday “ ({ 0s } ∪ ∅)) = ∅ → suc ∪ ( bday “ ({ 0s } ∪ ∅)) = suc ∅)
23	21, 22	ax-mp 5	. . . . 5 ⊢ suc ∪ ( bday “ ({ 0s } ∪ ∅)) = suc ∅
24		df-1o 8297	. . . . 5 ⊢ 1o = suc ∅
25	23, 24	eqtr4i 2769	. . . 4 ⊢ suc ∪ ( bday “ ({ 0s } ∪ ∅)) = 1o
26	9, 25	sseqtri 3957	. . 3 ⊢ ( bday ‘({ 0s } |s ∅)) ⊆ 1o
27		ssrab2 4013	. . . . . 6 ⊢ {𝑥 ∈ No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)} ⊆ No
28		fnssintima 33676	. . . . . 6 ⊢ (( bday Fn No ∧ {𝑥 ∈ No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)} ⊆ No ) → (1o ⊆ ∩ ( bday “ {𝑥 ∈ No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)}) ↔ ∀𝑦 ∈ {𝑥 ∈ No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)}1o ⊆ ( bday ‘𝑦)))
29	12, 27, 28	mp2an 689	. . . . 5 ⊢ (1o ⊆ ∩ ( bday “ {𝑥 ∈ No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)}) ↔ ∀𝑦 ∈ {𝑥 ∈ No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)}1o ⊆ ( bday ‘𝑦))
30		sneq 4571	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
31	30	breq2d 5086	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ({ 0s } <<s {𝑥} ↔ { 0s } <<s {𝑦}))
32	30	breq1d 5084	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ({𝑥} <<s ∅ ↔ {𝑦} <<s ∅))
33	31, 32	anbi12d 631	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅) ↔ ({ 0s } <<s {𝑦} ∧ {𝑦} <<s ∅)))
34	33	elrab 3624	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∈ No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)} ↔ (𝑦 ∈ No ∧ ({ 0s } <<s {𝑦} ∧ {𝑦} <<s ∅)))
35		sltirr 33949	. . . . . . . . . . . . 13 ⊢ ( 0s ∈ No → ¬ 0s <s 0s )
36	3, 35	ax-mp 5	. . . . . . . . . . . 12 ⊢ ¬ 0s <s 0s
37		breq2 5078	. . . . . . . . . . . 12 ⊢ (𝑦 = 0s → ( 0s <s 𝑦 ↔ 0s <s 0s ))
38	36, 37	mtbiri 327	. . . . . . . . . . 11 ⊢ (𝑦 = 0s → ¬ 0s <s 𝑦)
39	38	necon2ai 2973	. . . . . . . . . 10 ⊢ ( 0s <s 𝑦 → 𝑦 ≠ 0s )
40		bday0b 34024	. . . . . . . . . . 11 ⊢ (𝑦 ∈ No → (( bday ‘𝑦) = ∅ ↔ 𝑦 = 0s ))
41	40	necon3bid 2988	. . . . . . . . . 10 ⊢ (𝑦 ∈ No → (( bday ‘𝑦) ≠ ∅ ↔ 𝑦 ≠ 0s ))
42	39, 41	syl5ibr 245	. . . . . . . . 9 ⊢ (𝑦 ∈ No → ( 0s <s 𝑦 → ( bday ‘𝑦) ≠ ∅))
43		bdayelon 33971	. . . . . . . . . . 11 ⊢ ( bday ‘𝑦) ∈ On
44	43	onordi 6371	. . . . . . . . . 10 ⊢ Ord ( bday ‘𝑦)
45		ordge1n0 8324	. . . . . . . . . 10 ⊢ (Ord ( bday ‘𝑦) → (1o ⊆ ( bday ‘𝑦) ↔ ( bday ‘𝑦) ≠ ∅))
46	44, 45	ax-mp 5	. . . . . . . . 9 ⊢ (1o ⊆ ( bday ‘𝑦) ↔ ( bday ‘𝑦) ≠ ∅)
47	42, 46	syl6ibr 251	. . . . . . . 8 ⊢ (𝑦 ∈ No → ( 0s <s 𝑦 → 1o ⊆ ( bday ‘𝑦)))
48		ssltsep 33985	. . . . . . . . 9 ⊢ ({ 0s } <<s {𝑦} → ∀𝑥 ∈ { 0s }∀𝑧 ∈ {𝑦}𝑥 <s 𝑧)
49		vex 3436	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
50		breq2 5078	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑦 → (𝑥 <s 𝑧 ↔ 𝑥 <s 𝑦))
51	49, 50	ralsn 4617	. . . . . . . . . . 11 ⊢ (∀𝑧 ∈ {𝑦}𝑥 <s 𝑧 ↔ 𝑥 <s 𝑦)
52	51	ralbii 3092	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ { 0s }∀𝑧 ∈ {𝑦}𝑥 <s 𝑧 ↔ ∀𝑥 ∈ { 0s }𝑥 <s 𝑦)
53	3	elexi 3451	. . . . . . . . . . 11 ⊢ 0s ∈ V
54		breq1 5077	. . . . . . . . . . 11 ⊢ (𝑥 = 0s → (𝑥 <s 𝑦 ↔ 0s <s 𝑦))
55	53, 54	ralsn 4617	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ { 0s }𝑥 <s 𝑦 ↔ 0s <s 𝑦)
56	52, 55	bitri 274	. . . . . . . . 9 ⊢ (∀𝑥 ∈ { 0s }∀𝑧 ∈ {𝑦}𝑥 <s 𝑧 ↔ 0s <s 𝑦)
57	48, 56	sylib 217	. . . . . . . 8 ⊢ ({ 0s } <<s {𝑦} → 0s <s 𝑦)
58	47, 57	impel 506	. . . . . . 7 ⊢ ((𝑦 ∈ No ∧ { 0s } <<s {𝑦}) → 1o ⊆ ( bday ‘𝑦))
59	58	adantrr 714	. . . . . 6 ⊢ ((𝑦 ∈ No ∧ ({ 0s } <<s {𝑦} ∧ {𝑦} <<s ∅)) → 1o ⊆ ( bday ‘𝑦))
60	34, 59	sylbi 216	. . . . 5 ⊢ (𝑦 ∈ {𝑥 ∈ No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)} → 1o ⊆ ( bday ‘𝑦))
61	29, 60	mprgbir 3079	. . . 4 ⊢ 1o ⊆ ∩ ( bday “ {𝑥 ∈ No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)})
62		scutbday 33998	. . . . 5 ⊢ ({ 0s } <<s ∅ → ( bday ‘({ 0s } |s ∅)) = ∩ ( bday “ {𝑥 ∈ No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)}))
63	7, 62	ax-mp 5	. . . 4 ⊢ ( bday ‘({ 0s } |s ∅)) = ∩ ( bday “ {𝑥 ∈ No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)})
64	61, 63	sseqtrri 3958	. . 3 ⊢ 1o ⊆ ( bday ‘({ 0s } |s ∅))
65	26, 64	eqssi 3937	. 2 ⊢ ( bday ‘({ 0s } |s ∅)) = 1o
66	2, 65	eqtri 2766	1 ⊢ ( bday ‘ 1s ) = 1o

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  {crab 3068   ∪ cun 3885   ⊆ wss 3887  ∅c0 4256  𝒫 cpw 4533  {csn 4561  ∪ cuni 4839  ∩ cint 4879   class class class wbr 5074   “ cima 5592  Ord word 6265  suc csuc 6268   Fn wfn 6428  ‘cfv 6433  (class class class)co 7275  1_oc1o 8290   No csur 33843   <s cslt 33844   bday cbday 33845   <<s csslt 33975   |s cscut 33977   0s c0s 34016   1s c1s 34017

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1o 8297  df-2o 8298  df-no 33846  df-slt 33847  df-bday 33848  df-sslt 33976  df-scut 33978  df-0s 34018  df-1s 34019

This theorem is referenced by: (None)