Theorem bday1 28233

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 28227	. . 3 ⊢ 1s = ({ 0s } |s ∅)
2	1	fveq2i 6914	. 2 ⊢ (bday‘ 1s ) = (bday‘({ 0s } |s ∅))
3		0no 28228	. . . . . . 7 ⊢ 0s ∈ No
4		snelpwi 5417	. . . . . . 7 ⊢ ( 0s ∈ No → { 0s } ∈ 𝒫 No)
5	3, 4	ax-mp 5	. . . . . 6 ⊢ { 0s } ∈ 𝒫 No
6		nulsgts 28195	. . . . . 6 ⊢ ({ 0s } ∈ 𝒫 No → { 0s } <<s ∅)
7	5, 6	ax-mp 5	. . . . 5 ⊢ { 0s } <<s ∅
8		cutbdaybnd2 28215	. . . . 5 ⊢ ({ 0s } <<s ∅ → (bday‘({ 0s } |s ∅)) ⊆ suc ∪ (bday “ ({ 0s } ∪ ∅)))
9	7, 8	ax-mp 5	. . . 4 ⊢ (bday‘({ 0s } |s ∅)) ⊆ suc ∪ (bday “ ({ 0s } ∪ ∅))
10		un0 4347	. . . . . . . . . 10 ⊢ ({ 0s } ∪ ∅) = { 0s }
11	10	imaeq2i 6063	. . . . . . . . 9 ⊢ (bday “ ({ 0s } ∪ ∅)) = (bday “ { 0s })
12		bdayfn 28167	. . . . . . . . . 10 ⊢ bday Fn No
13		fnsnfv 6990	. . . . . . . . . 10 ⊢ ((bday Fn No ∧ 0s ∈ No) → {(bday‘ 0s )} = (bday “ { 0s }))
14	12, 3, 13	mp2an 705	. . . . . . . . 9 ⊢ {(bday‘ 0s )} = (bday “ { 0s })
15		bday0 28230	. . . . . . . . . 10 ⊢ (bday‘ 0s ) = ∅
16	15	sneqi 4595	. . . . . . . . 9 ⊢ {(bday‘ 0s )} = {∅}
17	11, 14, 16	3eqtr2i 2792	. . . . . . . 8 ⊢ (bday “ ({ 0s } ∪ ∅)) = {∅}
18	17	unieqi 4882	. . . . . . 7 ⊢ ∪ (bday “ ({ 0s } ∪ ∅)) = ∪ {∅}
19		0ex 5265	. . . . . . . 8 ⊢ ∅ ∈ V
20	19	unisn 4889	. . . . . . 7 ⊢ ∪ {∅} = ∅
21	18, 20	eqtri 2786	. . . . . 6 ⊢ ∪ (bday “ ({ 0s } ∪ ∅)) = ∅
22		suceq 6452	. . . . . 6 ⊢ (∪ (bday “ ({ 0s } ∪ ∅)) = ∅ → suc ∪ (bday “ ({ 0s } ∪ ∅)) = suc ∅)
23	21, 22	ax-mp 5	. . . . 5 ⊢ suc ∪ (bday “ ({ 0s } ∪ ∅)) = suc ∅
24		df-1o 8507	. . . . 5 ⊢ 1o = suc ∅
25	23, 24	eqtr4i 2789	. . . 4 ⊢ suc ∪ (bday “ ({ 0s } ∪ ∅)) = 1o
26	9, 25	sseqtri 3961	. . 3 ⊢ (bday‘({ 0s } |s ∅)) ⊆ 1o
27		ssrab2 4008	. . . . . 6 ⊢ {𝑥 ∈ No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)} ⊆ No
28		fnssintima 7398	. . . . . 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 4594	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
31	30	breq2d 5118	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ({ 0s } <<s {𝑥} ↔ { 0s } <<s {𝑦}))
32	30	breq1d 5116	. . . . . . . 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 3648	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∈ No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)} ↔ (𝑦 ∈ No ∧ ({ 0s } <<s {𝑦} ∧ {𝑦} <<s ∅)))
35		ltsirr 28136	. . . . . . . . . . . . 13 ⊢ ( 0s ∈ No → ¬ 0s <s 0s )
36	3, 35	ax-mp 5	. . . . . . . . . . . 12 ⊢ ¬ 0s <s 0s
37		breq2 5110	. . . . . . . . . . . 12 ⊢ (𝑦 = 0s → ( 0s <s 𝑦 ↔ 0s <s 0s ))
38	36, 37	mtbiri 330	. . . . . . . . . . 11 ⊢ (𝑦 = 0s → ¬ 0s <s 𝑦)
39	38	necon2ai 2988	. . . . . . . . . 10 ⊢ ( 0s <s 𝑦 → 𝑦 ≠ 0s )
40		bday0b 28232	. . . . . . . . . . 11 ⊢ (𝑦 ∈ No → ((bday‘𝑦) = ∅ ↔ 𝑦 = 0s ))
41	40	necon3bid 3003	. . . . . . . . . 10 ⊢ (𝑦 ∈ No → ((bday‘𝑦) ≠ ∅ ↔ 𝑦 ≠ 0s ))
42	39, 41	imbitrrid 249	. . . . . . . . 9 ⊢ (𝑦 ∈ No → ( 0s <s 𝑦 → (bday‘𝑦) ≠ ∅))
43		bdayon 28171	. . . . . . . . . . 11 ⊢ (bday‘𝑦) ∈ On
44	43	onordi 6497	. . . . . . . . . 10 ⊢ Ord (bday‘𝑦)
45		ordge1n0 8533	. . . . . . . . . 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 28186	. . . . . . . . 9 ⊢ ({ 0s } <<s {𝑦} → ∀𝑥 ∈ { 0s }∀𝑧 ∈ {𝑦}𝑥 <s 𝑧)
49		vex 3458	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
50		breq2 5110	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑦 → (𝑥 <s 𝑧 ↔ 𝑥 <s 𝑦))
51	49, 50	ralsn 4642	. . . . . . . . . . 11 ⊢ (∀𝑧 ∈ {𝑦}𝑥 <s 𝑧 ↔ 𝑥 <s 𝑦)
52	51	ralbii 3112	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ { 0s }∀𝑧 ∈ {𝑦}𝑥 <s 𝑧 ↔ ∀𝑥 ∈ { 0s }𝑥 <s 𝑦)
53	3	elexi 3476	. . . . . . . . . . 11 ⊢ 0s ∈ V
54		breq1 5109	. . . . . . . . . . 11 ⊢ (𝑥 = 0s → (𝑥 <s 𝑦 ↔ 0s <s 𝑦))
55	53, 54	ralsn 4642	. . . . . . . . . 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 3087	. . . 4 ⊢ 1o ⊆ ∩ (bday “ {𝑥 ∈ No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)})
62		cutbday 28203	. . . . 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 3962	. . 3 ⊢ 1o ⊆ (bday‘({ 0s } |s ∅))
65	26, 64	eqssi 3929	. 2 ⊢ (bday‘({ 0s } |s ∅)) = 1o
66	2, 65	eqtri 2786	1 ⊢ (bday‘ 1s ) = 1o

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∀wral 3080  {crab 3416   ⊆ wss 3900   ∪ cun 4152  ∅c0 4281  𝒫 cpw 4560  {csn 4588  ∪ cuni 4870  ∩ cint 4910   class class class wbr 5106   “ cima 5659  Ord word 6382  suc csuc 6385   Fn wfn 6556  ‘cfv 6561  (class class class)co 7446  1_oc1o 8500  Nocsur 28030   <s clts 28031  bdaycbday 28032   <<s cslts 28176   |s ccuts 28178   0_s c0s 28224   1_s c1s 28225

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-ord 6386  df-on 6387  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7403  df-ov 7449  df-oprab 7450  df-mpo 7451  df-1o 8507  df-2o 8508  df-no 28033  df-lts 28034  df-bday 28035  df-slts 28177  df-cuts 28179  df-0s 28226  df-1s 28227

This theorem is used by:  cuteq1  28236  left1s  28314  right1s  28315  bdaypw2n0bndlem  28882