Theorem bday1s 27758

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 27752	. . 3 ⊢ 1s = ({ 0s } |s ∅)
2	1	fveq2i 6895	. 2 ⊢ ( bday ‘ 1s ) = ( bday ‘({ 0s } |s ∅))
3		0sno 27753	. . . . . . 7 ⊢ 0s ∈ No
4		snelpwi 5440	. . . . . . 7 ⊢ ( 0s ∈ No → { 0s } ∈ 𝒫 No )
5	3, 4	ax-mp 5	. . . . . 6 ⊢ { 0s } ∈ 𝒫 No
6		nulssgt 27725	. . . . . 6 ⊢ ({ 0s } ∈ 𝒫 No → { 0s } <<s ∅)
7	5, 6	ax-mp 5	. . . . 5 ⊢ { 0s } <<s ∅
8		scutbdaybnd2 27743	. . . . 5 ⊢ ({ 0s } <<s ∅ → ( bday ‘({ 0s } |s ∅)) ⊆ suc ∪ ( bday “ ({ 0s } ∪ ∅)))
9	7, 8	ax-mp 5	. . . 4 ⊢ ( bday ‘({ 0s } |s ∅)) ⊆ suc ∪ ( bday “ ({ 0s } ∪ ∅))
10		un0 4387	. . . . . . . . . 10 ⊢ ({ 0s } ∪ ∅) = { 0s }
11	10	imaeq2i 6056	. . . . . . . . 9 ⊢ ( bday “ ({ 0s } ∪ ∅)) = ( bday “ { 0s })
12		bdayfn 27700	. . . . . . . . . 10 ⊢ bday Fn No
13		fnsnfv 6972	. . . . . . . . . 10 ⊢ (( bday Fn No ∧ 0s ∈ No ) → {( bday ‘ 0s )} = ( bday “ { 0s }))
14	12, 3, 13	mp2an 691	. . . . . . . . 9 ⊢ {( bday ‘ 0s )} = ( bday “ { 0s })
15		bday0s 27755	. . . . . . . . . 10 ⊢ ( bday ‘ 0s ) = ∅
16	15	sneqi 4636	. . . . . . . . 9 ⊢ {( bday ‘ 0s )} = {∅}
17	11, 14, 16	3eqtr2i 2762	. . . . . . . 8 ⊢ ( bday “ ({ 0s } ∪ ∅)) = {∅}
18	17	unieqi 4916	. . . . . . 7 ⊢ ∪ ( bday “ ({ 0s } ∪ ∅)) = ∪ {∅}
19		0ex 5302	. . . . . . . 8 ⊢ ∅ ∈ V
20	19	unisn 4925	. . . . . . 7 ⊢ ∪ {∅} = ∅
21	18, 20	eqtri 2756	. . . . . 6 ⊢ ∪ ( bday “ ({ 0s } ∪ ∅)) = ∅
22		suceq 6430	. . . . . 6 ⊢ (∪ ( bday “ ({ 0s } ∪ ∅)) = ∅ → suc ∪ ( bday “ ({ 0s } ∪ ∅)) = suc ∅)
23	21, 22	ax-mp 5	. . . . 5 ⊢ suc ∪ ( bday “ ({ 0s } ∪ ∅)) = suc ∅
24		df-1o 8481	. . . . 5 ⊢ 1o = suc ∅
25	23, 24	eqtr4i 2759	. . . 4 ⊢ suc ∪ ( bday “ ({ 0s } ∪ ∅)) = 1o
26	9, 25	sseqtri 4015	. . 3 ⊢ ( bday ‘({ 0s } |s ∅)) ⊆ 1o
27		ssrab2 4074	. . . . . 6 ⊢ {𝑥 ∈ No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)} ⊆ No
28		fnssintima 7365	. . . . . 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 691	. . . . 5 ⊢ (1o ⊆ ∩ ( bday “ {𝑥 ∈ No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)}) ↔ ∀𝑦 ∈ {𝑥 ∈ No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)}1o ⊆ ( bday ‘𝑦))
30		sneq 4635	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
31	30	breq2d 5155	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ({ 0s } <<s {𝑥} ↔ { 0s } <<s {𝑦}))
32	30	breq1d 5153	. . . . . . . 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 3681	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∈ No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)} ↔ (𝑦 ∈ No ∧ ({ 0s } <<s {𝑦} ∧ {𝑦} <<s ∅)))
35		sltirr 27673	. . . . . . . . . . . . 13 ⊢ ( 0s ∈ No → ¬ 0s <s 0s )
36	3, 35	ax-mp 5	. . . . . . . . . . . 12 ⊢ ¬ 0s <s 0s
37		breq2 5147	. . . . . . . . . . . 12 ⊢ (𝑦 = 0s → ( 0s <s 𝑦 ↔ 0s <s 0s ))
38	36, 37	mtbiri 327	. . . . . . . . . . 11 ⊢ (𝑦 = 0s → ¬ 0s <s 𝑦)
39	38	necon2ai 2966	. . . . . . . . . 10 ⊢ ( 0s <s 𝑦 → 𝑦 ≠ 0s )
40		bday0b 27757	. . . . . . . . . . 11 ⊢ (𝑦 ∈ No → (( bday ‘𝑦) = ∅ ↔ 𝑦 = 0s ))
41	40	necon3bid 2981	. . . . . . . . . 10 ⊢ (𝑦 ∈ No → (( bday ‘𝑦) ≠ ∅ ↔ 𝑦 ≠ 0s ))
42	39, 41	imbitrrid 245	. . . . . . . . 9 ⊢ (𝑦 ∈ No → ( 0s <s 𝑦 → ( bday ‘𝑦) ≠ ∅))
43		bdayelon 27703	. . . . . . . . . . 11 ⊢ ( bday ‘𝑦) ∈ On
44	43	onordi 6475	. . . . . . . . . 10 ⊢ Ord ( bday ‘𝑦)
45		ordge1n0 8509	. . . . . . . . . 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 251	. . . . . . . 8 ⊢ (𝑦 ∈ No → ( 0s <s 𝑦 → 1o ⊆ ( bday ‘𝑦)))
48		ssltsep 27717	. . . . . . . . 9 ⊢ ({ 0s } <<s {𝑦} → ∀𝑥 ∈ { 0s }∀𝑧 ∈ {𝑦}𝑥 <s 𝑧)
49		vex 3474	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
50		breq2 5147	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑦 → (𝑥 <s 𝑧 ↔ 𝑥 <s 𝑦))
51	49, 50	ralsn 4682	. . . . . . . . . . 11 ⊢ (∀𝑧 ∈ {𝑦}𝑥 <s 𝑧 ↔ 𝑥 <s 𝑦)
52	51	ralbii 3089	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ { 0s }∀𝑧 ∈ {𝑦}𝑥 <s 𝑧 ↔ ∀𝑥 ∈ { 0s }𝑥 <s 𝑦)
53	3	elexi 3490	. . . . . . . . . . 11 ⊢ 0s ∈ V
54		breq1 5146	. . . . . . . . . . 11 ⊢ (𝑥 = 0s → (𝑥 <s 𝑦 ↔ 0s <s 𝑦))
55	53, 54	ralsn 4682	. . . . . . . . . 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 217	. . . . . . . 8 ⊢ ({ 0s } <<s {𝑦} → 0s <s 𝑦)
58	47, 57	impel 505	. . . . . . 7 ⊢ ((𝑦 ∈ No ∧ { 0s } <<s {𝑦}) → 1o ⊆ ( bday ‘𝑦))
59	58	adantrr 716	. . . . . 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 3064	. . . 4 ⊢ 1o ⊆ ∩ ( bday “ {𝑥 ∈ No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)})
62		scutbday 27731	. . . . 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 4016	. . 3 ⊢ 1o ⊆ ( bday ‘({ 0s } |s ∅))
65	26, 64	eqssi 3995	. 2 ⊢ ( bday ‘({ 0s } |s ∅)) = 1o
66	2, 65	eqtri 2756	1 ⊢ ( bday ‘ 1s ) = 1o

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099   ≠ wne 2936  ∀wral 3057  {crab 3428   ∪ cun 3943   ⊆ wss 3945  ∅c0 4319  𝒫 cpw 4599  {csn 4625  ∪ cuni 4904  ∩ cint 4945   class class class wbr 5143   “ cima 5676  Ord word 6363  suc csuc 6366   Fn wfn 6538  ‘cfv 6543  (class class class)co 7415  1_oc1o 8474   No csur 27567   <s cslt 27568   bday cbday 27569   <<s csslt 27707   |s cscut 27709   0_s c0s 27749   1_s c1s 27750

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pr 5424  ax-un 7735

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4905  df-int 4946  df-iun 4994  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-ord 6367  df-on 6368  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7371  df-ov 7418  df-oprab 7419  df-mpo 7420  df-1o 8481  df-2o 8482  df-no 27570  df-slt 27571  df-bday 27572  df-sslt 27708  df-scut 27710  df-0s 27751  df-1s 27752

This theorem is referenced by:  cuteq1  27760  left1s  27815  right1s  27816

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