Theorem bday1 27818

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 27812	. . 3 ⊢ 1s = ({ 0s } |s ∅)
2	1	fveq2i 6835	. 2 ⊢ ( bday ‘ 1s ) = ( bday ‘({ 0s } |s ∅))
3		0no 27813	. . . . . . 7 ⊢ 0s ∈ No
4		snelpwi 5389	. . . . . . 7 ⊢ ( 0s ∈ No → { 0s } ∈ 𝒫 No )
5	3, 4	ax-mp 5	. . . . . 6 ⊢ { 0s } ∈ 𝒫 No
6		nulsgts 27780	. . . . . 6 ⊢ ({ 0s } ∈ 𝒫 No → { 0s } <<s ∅)
7	5, 6	ax-mp 5	. . . . 5 ⊢ { 0s } <<s ∅
8		cutbdaybnd2 27800	. . . . 5 ⊢ ({ 0s } <<s ∅ → ( bday ‘({ 0s } |s ∅)) ⊆ suc ∪ ( bday “ ({ 0s } ∪ ∅)))
9	7, 8	ax-mp 5	. . . 4 ⊢ ( bday ‘({ 0s } |s ∅)) ⊆ suc ∪ ( bday “ ({ 0s } ∪ ∅))
10		un0 4335	. . . . . . . . . 10 ⊢ ({ 0s } ∪ ∅) = { 0s }
11	10	imaeq2i 6015	. . . . . . . . 9 ⊢ ( bday “ ({ 0s } ∪ ∅)) = ( bday “ { 0s })
12		bdayfn 27753	. . . . . . . . . 10 ⊢ bday Fn No
13		fnsnfv 6911	. . . . . . . . . 10 ⊢ (( bday Fn No ∧ 0s ∈ No ) → {( bday ‘ 0s )} = ( bday “ { 0s }))
14	12, 3, 13	mp2an 693	. . . . . . . . 9 ⊢ {( bday ‘ 0s )} = ( bday “ { 0s })
15		bday0 27815	. . . . . . . . . 10 ⊢ ( bday ‘ 0s ) = ∅
16	15	sneqi 4579	. . . . . . . . 9 ⊢ {( bday ‘ 0s )} = {∅}
17	11, 14, 16	3eqtr2i 2766	. . . . . . . 8 ⊢ ( bday “ ({ 0s } ∪ ∅)) = {∅}
18	17	unieqi 4863	. . . . . . 7 ⊢ ∪ ( bday “ ({ 0s } ∪ ∅)) = ∪ {∅}
19		0ex 5242	. . . . . . . 8 ⊢ ∅ ∈ V
20	19	unisn 4870	. . . . . . 7 ⊢ ∪ {∅} = ∅
21	18, 20	eqtri 2760	. . . . . 6 ⊢ ∪ ( bday “ ({ 0s } ∪ ∅)) = ∅
22		suceq 6383	. . . . . 6 ⊢ (∪ ( bday “ ({ 0s } ∪ ∅)) = ∅ → suc ∪ ( bday “ ({ 0s } ∪ ∅)) = suc ∅)
23	21, 22	ax-mp 5	. . . . 5 ⊢ suc ∪ ( bday “ ({ 0s } ∪ ∅)) = suc ∅
24		df-1o 8396	. . . . 5 ⊢ 1o = suc ∅
25	23, 24	eqtr4i 2763	. . . 4 ⊢ suc ∪ ( bday “ ({ 0s } ∪ ∅)) = 1o
26	9, 25	sseqtri 3971	. . 3 ⊢ ( bday ‘({ 0s } |s ∅)) ⊆ 1o
27		ssrab2 4021	. . . . . 6 ⊢ {𝑥 ∈ No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)} ⊆ No
28		fnssintima 7308	. . . . . 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 693	. . . . 5 ⊢ (1o ⊆ ∩ ( bday “ {𝑥 ∈ No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)}) ↔ ∀𝑦 ∈ {𝑥 ∈ No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)}1o ⊆ ( bday ‘𝑦))
30		sneq 4578	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
31	30	breq2d 5098	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ({ 0s } <<s {𝑥} ↔ { 0s } <<s {𝑦}))
32	30	breq1d 5096	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ({𝑥} <<s ∅ ↔ {𝑦} <<s ∅))
33	31, 32	anbi12d 633	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅) ↔ ({ 0s } <<s {𝑦} ∧ {𝑦} <<s ∅)))
34	33	elrab 3635	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∈ No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)} ↔ (𝑦 ∈ No ∧ ({ 0s } <<s {𝑦} ∧ {𝑦} <<s ∅)))
35		ltsirr 27722	. . . . . . . . . . . . 13 ⊢ ( 0s ∈ No → ¬ 0s <s 0s )
36	3, 35	ax-mp 5	. . . . . . . . . . . 12 ⊢ ¬ 0s <s 0s
37		breq2 5090	. . . . . . . . . . . 12 ⊢ (𝑦 = 0s → ( 0s <s 𝑦 ↔ 0s <s 0s ))
38	36, 37	mtbiri 327	. . . . . . . . . . 11 ⊢ (𝑦 = 0s → ¬ 0s <s 𝑦)
39	38	necon2ai 2962	. . . . . . . . . 10 ⊢ ( 0s <s 𝑦 → 𝑦 ≠ 0s )
40		bday0b 27817	. . . . . . . . . . 11 ⊢ (𝑦 ∈ No → (( bday ‘𝑦) = ∅ ↔ 𝑦 = 0s ))
41	40	necon3bid 2977	. . . . . . . . . 10 ⊢ (𝑦 ∈ No → (( bday ‘𝑦) ≠ ∅ ↔ 𝑦 ≠ 0s ))
42	39, 41	imbitrrid 246	. . . . . . . . 9 ⊢ (𝑦 ∈ No → ( 0s <s 𝑦 → ( bday ‘𝑦) ≠ ∅))
43		bdayon 27756	. . . . . . . . . . 11 ⊢ ( bday ‘𝑦) ∈ On
44	43	onordi 6428	. . . . . . . . . 10 ⊢ Ord ( bday ‘𝑦)
45		ordge1n0 8420	. . . . . . . . . 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 252	. . . . . . . 8 ⊢ (𝑦 ∈ No → ( 0s <s 𝑦 → 1o ⊆ ( bday ‘𝑦)))
48		sltssep 27771	. . . . . . . . 9 ⊢ ({ 0s } <<s {𝑦} → ∀𝑥 ∈ { 0s }∀𝑧 ∈ {𝑦}𝑥 <s 𝑧)
49		vex 3434	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
50		breq2 5090	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑦 → (𝑥 <s 𝑧 ↔ 𝑥 <s 𝑦))
51	49, 50	ralsn 4626	. . . . . . . . . . 11 ⊢ (∀𝑧 ∈ {𝑦}𝑥 <s 𝑧 ↔ 𝑥 <s 𝑦)
52	51	ralbii 3084	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ { 0s }∀𝑧 ∈ {𝑦}𝑥 <s 𝑧 ↔ ∀𝑥 ∈ { 0s }𝑥 <s 𝑦)
53	3	elexi 3453	. . . . . . . . . . 11 ⊢ 0s ∈ V
54		breq1 5089	. . . . . . . . . . 11 ⊢ (𝑥 = 0s → (𝑥 <s 𝑦 ↔ 0s <s 𝑦))
55	53, 54	ralsn 4626	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ { 0s }𝑥 <s 𝑦 ↔ 0s <s 𝑦)
56	52, 55	bitri 275	. . . . . . . . 9 ⊢ (∀𝑥 ∈ { 0s }∀𝑧 ∈ {𝑦}𝑥 <s 𝑧 ↔ 0s <s 𝑦)
57	48, 56	sylib 218	. . . . . . . 8 ⊢ ({ 0s } <<s {𝑦} → 0s <s 𝑦)
58	47, 57	impel 505	. . . . . . 7 ⊢ ((𝑦 ∈ No ∧ { 0s } <<s {𝑦}) → 1o ⊆ ( bday ‘𝑦))
59	58	adantrr 718	. . . . . 6 ⊢ ((𝑦 ∈ No ∧ ({ 0s } <<s {𝑦} ∧ {𝑦} <<s ∅)) → 1o ⊆ ( bday ‘𝑦))
60	34, 59	sylbi 217	. . . . 5 ⊢ (𝑦 ∈ {𝑥 ∈ No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)} → 1o ⊆ ( bday ‘𝑦))
61	29, 60	mprgbir 3059	. . . 4 ⊢ 1o ⊆ ∩ ( bday “ {𝑥 ∈ No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)})
62		cutbday 27788	. . . . 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 3972	. . 3 ⊢ 1o ⊆ ( bday ‘({ 0s } |s ∅))
65	26, 64	eqssi 3939	. 2 ⊢ ( bday ‘({ 0s } |s ∅)) = 1o
66	2, 65	eqtri 2760	1 ⊢ ( bday ‘ 1s ) = 1o

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  {crab 3390   ∪ cun 3888   ⊆ wss 3890  ∅c0 4274  𝒫 cpw 4542  {csn 4568  ∪ cuni 4851  ∩ cint 4890   class class class wbr 5086   “ cima 5625  Ord word 6314  suc csuc 6317   Fn wfn 6485  ‘cfv 6490  (class class class)co 7358  1_oc1o 8389   No csur 27615   <s clts 27616   bday cbday 27617   <<s cslts 27761   |s ccuts 27763   0_s c0s 27809   1_s c1s 27810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-int 4891  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-ord 6318  df-on 6319  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1o 8396  df-2o 8397  df-no 27618  df-lts 27619  df-bday 27620  df-slts 27762  df-cuts 27764  df-0s 27811  df-1s 27812

This theorem is referenced by:  cuteq1  27821  left1s  27899  right1s  27900  bdaypw2n0bndlem  28467