Theorem bday1 28205

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

Assertion

Ref	Expression
bday1	⊢ (bday‘ 1s ) = 1o

Proof of Theorem bday1

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

Step	Hyp	Ref	Expression
1		df-1s 28199	. . 3 ⊢ 1s = ({ 0s } |s ∅)
2	1	fveq2i 6888	. 2 ⊢ (bday‘ 1s ) = (bday‘({ 0s } |s ∅))
3		0no 28200	. . . . . . 7 ⊢ 0s ∈ No
4		snelpwi 5411	. . . . . . 7 ⊢ ( 0s ∈ No → { 0s } ∈ 𝒫 No)
5	3, 4	ax-mp 5	. . . . . 6 ⊢ { 0s } ∈ 𝒫 No
6		nulsgts 28167	. . . . . 6 ⊢ ({ 0s } ∈ 𝒫 No → { 0s } <<s ∅)
7	5, 6	ax-mp 5	. . . . 5 ⊢ { 0s } <<s ∅
8		cutbdaybnd2 28187	. . . . 5 ⊢ ({ 0s } <<s ∅ → (bday‘({ 0s } |s ∅)) ⊆ suc ∪ (bday “ ({ 0s } ∪ ∅)))
9	7, 8	ax-mp 5	. . . 4 ⊢ (bday‘({ 0s } |s ∅)) ⊆ suc ∪ (bday “ ({ 0s } ∪ ∅))
10		un0 4343	. . . . . . . . . 10 ⊢ ({ 0s } ∪ ∅) = { 0s }
11	10	imaeq2i 6049	. . . . . . . . 9 ⊢ (bday “ ({ 0s } ∪ ∅)) = (bday “ { 0s })
12		bdayfn 28139	. . . . . . . . . 10 ⊢ bday Fn No
13		fnsnfv 6964	. . . . . . . . . 10 ⊢ ((bday Fn No ∧ 0s ∈ No) → {(bday‘ 0s )} = (bday “ { 0s }))
14	12, 3, 13	mp2an 705	. . . . . . . . 9 ⊢ {(bday‘ 0s )} = (bday “ { 0s })
15		bday0 28202	. . . . . . . . . 10 ⊢ (bday‘ 0s ) = ∅
16	15	sneqi 4591	. . . . . . . . 9 ⊢ {(bday‘ 0s )} = {∅}
17	11, 14, 16	3eqtr2i 2789	. . . . . . . 8 ⊢ (bday “ ({ 0s } ∪ ∅)) = {∅}
18	17	unieqi 4878	. . . . . . 7 ⊢ ∪ (bday “ ({ 0s } ∪ ∅)) = ∪ {∅}
19		0ex 5260	. . . . . . . 8 ⊢ ∅ ∈ V
20	19	unisn 4885	. . . . . . 7 ⊢ ∪ {∅} = ∅
21	18, 20	eqtri 2783	. . . . . 6 ⊢ ∪ (bday “ ({ 0s } ∪ ∅)) = ∅
22		suceq 6431	. . . . . 6 ⊢ (∪ (bday “ ({ 0s } ∪ ∅)) = ∅ → suc ∪ (bday “ ({ 0s } ∪ ∅)) = suc ∅)
23	21, 22	ax-mp 5	. . . . 5 ⊢ suc ∪ (bday “ ({ 0s } ∪ ∅)) = suc ∅
24		df-1o 8479	. . . . 5 ⊢ 1o = suc ∅
25	23, 24	eqtr4i 2786	. . . 4 ⊢ suc ∪ (bday “ ({ 0s } ∪ ∅)) = 1o
26	9, 25	sseqtri 3956	. . 3 ⊢ (bday‘({ 0s } |s ∅)) ⊆ 1o
27		ssrab2 4005	. . . . . 6 ⊢ {𝑥 ∈ No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)} ⊆ No
28		fnssintima 7372	. . . . . 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 705	. . . . 5 ⊢ (1o ⊆ ∩ (bday “ {𝑥 ∈ No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)}) ↔ ∀𝑦 ∈ {𝑥 ∈ No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)}1o ⊆ (bday‘𝑦))
30		sneq 4590	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
31	30	breq2d 5114	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ({ 0s } <<s {𝑥} ↔ { 0s } <<s {𝑦}))
32	30	breq1d 5112	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ({𝑥} <<s ∅ ↔ {𝑦} <<s ∅))
33	31, 32	anbi12d 644	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅) ↔ ({ 0s } <<s {𝑦} ∧ {𝑦} <<s ∅)))
34	33	elrab 3644	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∈ No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)} ↔ (𝑦 ∈ No ∧ ({ 0s } <<s {𝑦} ∧ {𝑦} <<s ∅)))
35		ltsirr 28108	. . . . . . . . . . . . 13 ⊢ ( 0s ∈ No → ¬ 0s <s 0s )
36	3, 35	ax-mp 5	. . . . . . . . . . . 12 ⊢ ¬ 0s <s 0s
37		breq2 5106	. . . . . . . . . . . 12 ⊢ (𝑦 = 0s → ( 0s <s 𝑦 ↔ 0s <s 0s ))
38	36, 37	mtbiri 330	. . . . . . . . . . 11 ⊢ (𝑦 = 0s → ¬ 0s <s 𝑦)
39	38	necon2ai 2984	. . . . . . . . . 10 ⊢ ( 0s <s 𝑦 → 𝑦 ≠ 0s )
40		bday0b 28204	. . . . . . . . . . 11 ⊢ (𝑦 ∈ No → ((bday‘𝑦) = ∅ ↔ 𝑦 = 0s ))
41	40	necon3bid 2999	. . . . . . . . . 10 ⊢ (𝑦 ∈ No → ((bday‘𝑦) ≠ ∅ ↔ 𝑦 ≠ 0s ))
42	39, 41	imbitrrid 249	. . . . . . . . 9 ⊢ (𝑦 ∈ No → ( 0s <s 𝑦 → (bday‘𝑦) ≠ ∅))
43		bdayon 28143	. . . . . . . . . . 11 ⊢ (bday‘𝑦) ∈ On
44	43	onordi 6476	. . . . . . . . . 10 ⊢ Ord (bday‘𝑦)
45		ordge1n0 8505	. . . . . . . . . 10 ⊢ (Ord (bday‘𝑦) → (1o ⊆ (bday‘𝑦) ↔ (bday‘𝑦) ≠ ∅))
46	44, 45	ax-mp 5	. . . . . . . . 9 ⊢ (1o ⊆ (bday‘𝑦) ↔ (bday‘𝑦) ≠ ∅)
47	42, 46	imbitrrdi 255	. . . . . . . 8 ⊢ (𝑦 ∈ No → ( 0s <s 𝑦 → 1o ⊆ (bday‘𝑦)))
48		sltssep 28158	. . . . . . . . 9 ⊢ ({ 0s } <<s {𝑦} → ∀𝑥 ∈ { 0s }∀𝑧 ∈ {𝑦}𝑥 <s 𝑧)
49		vex 3454	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
50		breq2 5106	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑦 → (𝑥 <s 𝑧 ↔ 𝑥 <s 𝑦))
51	49, 50	ralsn 4638	. . . . . . . . . . 11 ⊢ (∀𝑧 ∈ {𝑦}𝑥 <s 𝑧 ↔ 𝑥 <s 𝑦)
52	51	ralbii 3108	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ { 0s }∀𝑧 ∈ {𝑦}𝑥 <s 𝑧 ↔ ∀𝑥 ∈ { 0s }𝑥 <s 𝑦)
53	3	elexi 3472	. . . . . . . . . . 11 ⊢ 0s ∈ V
54		breq1 5105	. . . . . . . . . . 11 ⊢ (𝑥 = 0s → (𝑥 <s 𝑦 ↔ 0s <s 𝑦))
55	53, 54	ralsn 4638	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ { 0s }𝑥 <s 𝑦 ↔ 0s <s 𝑦)
56	52, 55	bitri 278	. . . . . . . . 9 ⊢ (∀𝑥 ∈ { 0s }∀𝑧 ∈ {𝑦}𝑥 <s 𝑧 ↔ 0s <s 𝑦)
57	48, 56	sylib 221	. . . . . . . 8 ⊢ ({ 0s } <<s {𝑦} → 0s <s 𝑦)
58	47, 57	impel 515	. . . . . . 7 ⊢ ((𝑦 ∈ No ∧ { 0s } <<s {𝑦}) → 1o ⊆ (bday‘𝑦))
59	58	adantrr 730	. . . . . 6 ⊢ ((𝑦 ∈ No ∧ ({ 0s } <<s {𝑦} ∧ {𝑦} <<s ∅)) → 1o ⊆ (bday‘𝑦))
60	34, 59	sylbi 220	. . . . 5 ⊢ (𝑦 ∈ {𝑥 ∈ No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)} → 1o ⊆ (bday‘𝑦))
61	29, 60	mprgbir 3083	. . . 4 ⊢ 1o ⊆ ∩ (bday “ {𝑥 ∈ No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)})
62		cutbday 28175	. . . . 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 3957	. . 3 ⊢ 1o ⊆ (bday‘({ 0s } |s ∅))
65	26, 64	eqssi 3924	. 2 ⊢ (bday‘({ 0s } |s ∅)) = 1o
66	2, 65	eqtri 2783	1 ⊢ (bday‘ 1s ) = 1o

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  {crab 3412   ⊆ wss 3895   ∪ cun 4149  ∅c0 4278  𝒫 cpw 4556  {csn 4584  ∪ cuni 4866  ∩ cint 4906   class class class wbr 5102   “ cima 5653  Ord word 6361  suc csuc 6364   Fn wfn 6533  ‘cfv 6538  (class class class)co 7420  1_oc1o 8472  Nocsur 28002   <s clts 28003  bdaycbday 28004   <<s cslts 28148   |s ccuts 28150   0_s c0s 28196   1_s c1s 28197

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 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-int 4907  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-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-ord 6365  df-on 6366  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-1o 8479  df-2o 8480  df-no 28005  df-lts 28006  df-bday 28007  df-slts 28149  df-cuts 28151  df-0s 28198  df-1s 28199

This theorem is used by:  cuteq1  28208  left1s  28286  right1s  28287  bdaypw2n0bndlem  28854