Theorem bday1s 27894

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 27888	. . 3 ⊢ 1s = ({ 0s } |s ∅)
2	1	fveq2i 6923	. 2 ⊢ ( bday ‘ 1s ) = ( bday ‘({ 0s } |s ∅))
3		0sno 27889	. . . . . . 7 ⊢ 0s ∈ No
4		snelpwi 5463	. . . . . . 7 ⊢ ( 0s ∈ No → { 0s } ∈ 𝒫 No )
5	3, 4	ax-mp 5	. . . . . 6 ⊢ { 0s } ∈ 𝒫 No
6		nulssgt 27861	. . . . . 6 ⊢ ({ 0s } ∈ 𝒫 No → { 0s } <<s ∅)
7	5, 6	ax-mp 5	. . . . 5 ⊢ { 0s } <<s ∅
8		scutbdaybnd2 27879	. . . . 5 ⊢ ({ 0s } <<s ∅ → ( bday ‘({ 0s } |s ∅)) ⊆ suc ∪ ( bday “ ({ 0s } ∪ ∅)))
9	7, 8	ax-mp 5	. . . 4 ⊢ ( bday ‘({ 0s } |s ∅)) ⊆ suc ∪ ( bday “ ({ 0s } ∪ ∅))
10		un0 4417	. . . . . . . . . 10 ⊢ ({ 0s } ∪ ∅) = { 0s }
11	10	imaeq2i 6087	. . . . . . . . 9 ⊢ ( bday “ ({ 0s } ∪ ∅)) = ( bday “ { 0s })
12		bdayfn 27836	. . . . . . . . . 10 ⊢ bday Fn No
13		fnsnfv 7001	. . . . . . . . . 10 ⊢ (( bday Fn No ∧ 0s ∈ No ) → {( bday ‘ 0s )} = ( bday “ { 0s }))
14	12, 3, 13	mp2an 691	. . . . . . . . 9 ⊢ {( bday ‘ 0s )} = ( bday “ { 0s })
15		bday0s 27891	. . . . . . . . . 10 ⊢ ( bday ‘ 0s ) = ∅
16	15	sneqi 4659	. . . . . . . . 9 ⊢ {( bday ‘ 0s )} = {∅}
17	11, 14, 16	3eqtr2i 2774	. . . . . . . 8 ⊢ ( bday “ ({ 0s } ∪ ∅)) = {∅}
18	17	unieqi 4943	. . . . . . 7 ⊢ ∪ ( bday “ ({ 0s } ∪ ∅)) = ∪ {∅}
19		0ex 5325	. . . . . . . 8 ⊢ ∅ ∈ V
20	19	unisn 4950	. . . . . . 7 ⊢ ∪ {∅} = ∅
21	18, 20	eqtri 2768	. . . . . 6 ⊢ ∪ ( bday “ ({ 0s } ∪ ∅)) = ∅
22		suceq 6461	. . . . . 6 ⊢ (∪ ( bday “ ({ 0s } ∪ ∅)) = ∅ → suc ∪ ( bday “ ({ 0s } ∪ ∅)) = suc ∅)
23	21, 22	ax-mp 5	. . . . 5 ⊢ suc ∪ ( bday “ ({ 0s } ∪ ∅)) = suc ∅
24		df-1o 8522	. . . . 5 ⊢ 1o = suc ∅
25	23, 24	eqtr4i 2771	. . . 4 ⊢ suc ∪ ( bday “ ({ 0s } ∪ ∅)) = 1o
26	9, 25	sseqtri 4045	. . 3 ⊢ ( bday ‘({ 0s } |s ∅)) ⊆ 1o
27		ssrab2 4103	. . . . . 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 691	. . . . 5 ⊢ (1o ⊆ ∩ ( bday “ {𝑥 ∈ No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)}) ↔ ∀𝑦 ∈ {𝑥 ∈ No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)}1o ⊆ ( bday ‘𝑦))
30		sneq 4658	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
31	30	breq2d 5178	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ({ 0s } <<s {𝑥} ↔ { 0s } <<s {𝑦}))
32	30	breq1d 5176	. . . . . . . 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 3708	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∈ No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)} ↔ (𝑦 ∈ No ∧ ({ 0s } <<s {𝑦} ∧ {𝑦} <<s ∅)))
35		sltirr 27809	. . . . . . . . . . . . 13 ⊢ ( 0s ∈ No → ¬ 0s <s 0s )
36	3, 35	ax-mp 5	. . . . . . . . . . . 12 ⊢ ¬ 0s <s 0s
37		breq2 5170	. . . . . . . . . . . 12 ⊢ (𝑦 = 0s → ( 0s <s 𝑦 ↔ 0s <s 0s ))
38	36, 37	mtbiri 327	. . . . . . . . . . 11 ⊢ (𝑦 = 0s → ¬ 0s <s 𝑦)
39	38	necon2ai 2976	. . . . . . . . . 10 ⊢ ( 0s <s 𝑦 → 𝑦 ≠ 0s )
40		bday0b 27893	. . . . . . . . . . 11 ⊢ (𝑦 ∈ No → (( bday ‘𝑦) = ∅ ↔ 𝑦 = 0s ))
41	40	necon3bid 2991	. . . . . . . . . 10 ⊢ (𝑦 ∈ No → (( bday ‘𝑦) ≠ ∅ ↔ 𝑦 ≠ 0s ))
42	39, 41	imbitrrid 246	. . . . . . . . 9 ⊢ (𝑦 ∈ No → ( 0s <s 𝑦 → ( bday ‘𝑦) ≠ ∅))
43		bdayelon 27839	. . . . . . . . . . 11 ⊢ ( bday ‘𝑦) ∈ On
44	43	onordi 6506	. . . . . . . . . 10 ⊢ Ord ( bday ‘𝑦)
45		ordge1n0 8550	. . . . . . . . . 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		ssltsep 27853	. . . . . . . . 9 ⊢ ({ 0s } <<s {𝑦} → ∀𝑥 ∈ { 0s }∀𝑧 ∈ {𝑦}𝑥 <s 𝑧)
49		vex 3492	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
50		breq2 5170	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑦 → (𝑥 <s 𝑧 ↔ 𝑥 <s 𝑦))
51	49, 50	ralsn 4705	. . . . . . . . . . 11 ⊢ (∀𝑧 ∈ {𝑦}𝑥 <s 𝑧 ↔ 𝑥 <s 𝑦)
52	51	ralbii 3099	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ { 0s }∀𝑧 ∈ {𝑦}𝑥 <s 𝑧 ↔ ∀𝑥 ∈ { 0s }𝑥 <s 𝑦)
53	3	elexi 3511	. . . . . . . . . . 11 ⊢ 0s ∈ V
54		breq1 5169	. . . . . . . . . . 11 ⊢ (𝑥 = 0s → (𝑥 <s 𝑦 ↔ 0s <s 𝑦))
55	53, 54	ralsn 4705	. . . . . . . . . 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 716	. . . . . 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 3074	. . . 4 ⊢ 1o ⊆ ∩ ( bday “ {𝑥 ∈ No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)})
62		scutbday 27867	. . . . 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 4046	. . 3 ⊢ 1o ⊆ ( bday ‘({ 0s } |s ∅))
65	26, 64	eqssi 4025	. 2 ⊢ ( bday ‘({ 0s } |s ∅)) = 1o
66	2, 65	eqtri 2768	1 ⊢ ( bday ‘ 1s ) = 1o

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  {crab 3443   ∪ cun 3974   ⊆ wss 3976  ∅c0 4352  𝒫 cpw 4622  {csn 4648  ∪ cuni 4931  ∩ cint 4970   class class class wbr 5166   “ cima 5703  Ord word 6394  suc csuc 6397   Fn wfn 6568  ‘cfv 6573  (class class class)co 7448  1_oc1o 8515   No csur 27702   <s cslt 27703   bday cbday 27704   <<s csslt 27843   |s cscut 27845   0_s c0s 27885   1_s c1s 27886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1o 8522  df-2o 8523  df-no 27705  df-slt 27706  df-bday 27707  df-sslt 27844  df-scut 27846  df-0s 27887  df-1s 27888

This theorem is referenced by:  cuteq1  27896  left1s  27951  right1s  27952  pw2bday  28436