Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bday1s Structured version   Visualization version   GIF version

Theorem bday1s 34025
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.
StepHypRef Expression
1 df-1s 34019 . . 3 1s = ({ 0s } |s ∅)
21fveq2i 6777 . 2 ( bday ‘ 1s ) = ( bday ‘({ 0s } |s ∅))
3 0sno 34020 . . . . . . 7 0s ∈ No
4 snelpwi 5360 . . . . . . 7 ( 0s ∈ No → { 0s } ∈ 𝒫 No )
53, 4ax-mp 5 . . . . . 6 { 0s } ∈ 𝒫 No
6 nulssgt 33992 . . . . . 6 ({ 0s } ∈ 𝒫 No → { 0s } <<s ∅)
75, 6ax-mp 5 . . . . 5 { 0s } <<s ∅
8 scutbdaybnd2 34010 . . . . 5 ({ 0s } <<s ∅ → ( bday ‘({ 0s } |s ∅)) ⊆ suc ( bday “ ({ 0s } ∪ ∅)))
97, 8ax-mp 5 . . . 4 ( bday ‘({ 0s } |s ∅)) ⊆ suc ( bday “ ({ 0s } ∪ ∅))
10 un0 4324 . . . . . . . . . 10 ({ 0s } ∪ ∅) = { 0s }
1110imaeq2i 5967 . . . . . . . . 9 ( bday “ ({ 0s } ∪ ∅)) = ( bday “ { 0s })
12 bdayfn 33968 . . . . . . . . . 10 bday Fn No
13 fnsnfv 6847 . . . . . . . . . 10 (( bday Fn No ∧ 0s ∈ No ) → {( bday ‘ 0s )} = ( bday “ { 0s }))
1412, 3, 13mp2an 689 . . . . . . . . 9 {( bday ‘ 0s )} = ( bday “ { 0s })
15 bday0s 34022 . . . . . . . . . 10 ( bday ‘ 0s ) = ∅
1615sneqi 4572 . . . . . . . . 9 {( bday ‘ 0s )} = {∅}
1711, 14, 163eqtr2i 2772 . . . . . . . 8 ( bday “ ({ 0s } ∪ ∅)) = {∅}
1817unieqi 4852 . . . . . . 7 ( bday “ ({ 0s } ∪ ∅)) = {∅}
19 0ex 5231 . . . . . . . 8 ∅ ∈ V
2019unisn 4861 . . . . . . 7 {∅} = ∅
2118, 20eqtri 2766 . . . . . 6 ( bday “ ({ 0s } ∪ ∅)) = ∅
22 suceq 6331 . . . . . 6 ( ( bday “ ({ 0s } ∪ ∅)) = ∅ → suc ( bday “ ({ 0s } ∪ ∅)) = suc ∅)
2321, 22ax-mp 5 . . . . 5 suc ( bday “ ({ 0s } ∪ ∅)) = suc ∅
24 df-1o 8297 . . . . 5 1o = suc ∅
2523, 24eqtr4i 2769 . . . 4 suc ( bday “ ({ 0s } ∪ ∅)) = 1o
269, 25sseqtri 3957 . . 3 ( bday ‘({ 0s } |s ∅)) ⊆ 1o
27 ssrab2 4013 . . . . . 6 {𝑥 No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)} ⊆ No
28 fnssintima 33676 . . . . . 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)))
2912, 27, 28mp2an 689 . . . . 5 (1o ( bday “ {𝑥 No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)}) ↔ ∀𝑦 ∈ {𝑥 No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)}1o ⊆ ( bday 𝑦))
30 sneq 4571 . . . . . . . . 9 (𝑥 = 𝑦 → {𝑥} = {𝑦})
3130breq2d 5086 . . . . . . . 8 (𝑥 = 𝑦 → ({ 0s } <<s {𝑥} ↔ { 0s } <<s {𝑦}))
3230breq1d 5084 . . . . . . . 8 (𝑥 = 𝑦 → ({𝑥} <<s ∅ ↔ {𝑦} <<s ∅))
3331, 32anbi12d 631 . . . . . . 7 (𝑥 = 𝑦 → (({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅) ↔ ({ 0s } <<s {𝑦} ∧ {𝑦} <<s ∅)))
3433elrab 3624 . . . . . 6 (𝑦 ∈ {𝑥 No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)} ↔ (𝑦 No ∧ ({ 0s } <<s {𝑦} ∧ {𝑦} <<s ∅)))
35 sltirr 33949 . . . . . . . . . . . . 13 ( 0s ∈ No → ¬ 0s <s 0s )
363, 35ax-mp 5 . . . . . . . . . . . 12 ¬ 0s <s 0s
37 breq2 5078 . . . . . . . . . . . 12 (𝑦 = 0s → ( 0s <s 𝑦 ↔ 0s <s 0s ))
3836, 37mtbiri 327 . . . . . . . . . . 11 (𝑦 = 0s → ¬ 0s <s 𝑦)
3938necon2ai 2973 . . . . . . . . . 10 ( 0s <s 𝑦𝑦 ≠ 0s )
40 bday0b 34024 . . . . . . . . . . 11 (𝑦 No → (( bday 𝑦) = ∅ ↔ 𝑦 = 0s ))
4140necon3bid 2988 . . . . . . . . . 10 (𝑦 No → (( bday 𝑦) ≠ ∅ ↔ 𝑦 ≠ 0s ))
4239, 41syl5ibr 245 . . . . . . . . 9 (𝑦 No → ( 0s <s 𝑦 → ( bday 𝑦) ≠ ∅))
43 bdayelon 33971 . . . . . . . . . . 11 ( bday 𝑦) ∈ On
4443onordi 6371 . . . . . . . . . 10 Ord ( bday 𝑦)
45 ordge1n0 8324 . . . . . . . . . 10 (Ord ( bday 𝑦) → (1o ⊆ ( bday 𝑦) ↔ ( bday 𝑦) ≠ ∅))
4644, 45ax-mp 5 . . . . . . . . 9 (1o ⊆ ( bday 𝑦) ↔ ( bday 𝑦) ≠ ∅)
4742, 46syl6ibr 251 . . . . . . . 8 (𝑦 No → ( 0s <s 𝑦 → 1o ⊆ ( bday 𝑦)))
48 ssltsep 33985 . . . . . . . . 9 ({ 0s } <<s {𝑦} → ∀𝑥 ∈ { 0s }∀𝑧 ∈ {𝑦}𝑥 <s 𝑧)
49 vex 3436 . . . . . . . . . . . 12 𝑦 ∈ V
50 breq2 5078 . . . . . . . . . . . 12 (𝑧 = 𝑦 → (𝑥 <s 𝑧𝑥 <s 𝑦))
5149, 50ralsn 4617 . . . . . . . . . . 11 (∀𝑧 ∈ {𝑦}𝑥 <s 𝑧𝑥 <s 𝑦)
5251ralbii 3092 . . . . . . . . . 10 (∀𝑥 ∈ { 0s }∀𝑧 ∈ {𝑦}𝑥 <s 𝑧 ↔ ∀𝑥 ∈ { 0s }𝑥 <s 𝑦)
533elexi 3451 . . . . . . . . . . 11 0s ∈ V
54 breq1 5077 . . . . . . . . . . 11 (𝑥 = 0s → (𝑥 <s 𝑦 ↔ 0s <s 𝑦))
5553, 54ralsn 4617 . . . . . . . . . 10 (∀𝑥 ∈ { 0s }𝑥 <s 𝑦 ↔ 0s <s 𝑦)
5652, 55bitri 274 . . . . . . . . 9 (∀𝑥 ∈ { 0s }∀𝑧 ∈ {𝑦}𝑥 <s 𝑧 ↔ 0s <s 𝑦)
5748, 56sylib 217 . . . . . . . 8 ({ 0s } <<s {𝑦} → 0s <s 𝑦)
5847, 57impel 506 . . . . . . 7 ((𝑦 No ∧ { 0s } <<s {𝑦}) → 1o ⊆ ( bday 𝑦))
5958adantrr 714 . . . . . 6 ((𝑦 No ∧ ({ 0s } <<s {𝑦} ∧ {𝑦} <<s ∅)) → 1o ⊆ ( bday 𝑦))
6034, 59sylbi 216 . . . . 5 (𝑦 ∈ {𝑥 No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)} → 1o ⊆ ( bday 𝑦))
6129, 60mprgbir 3079 . . . 4 1o ( bday “ {𝑥 No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)})
62 scutbday 33998 . . . . 5 ({ 0s } <<s ∅ → ( bday ‘({ 0s } |s ∅)) = ( bday “ {𝑥 No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)}))
637, 62ax-mp 5 . . . 4 ( bday ‘({ 0s } |s ∅)) = ( bday “ {𝑥 No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)})
6461, 63sseqtrri 3958 . . 3 1o ⊆ ( bday ‘({ 0s } |s ∅))
6526, 64eqssi 3937 . 2 ( bday ‘({ 0s } |s ∅)) = 1o
662, 65eqtri 2766 1 ( bday ‘ 1s ) = 1o
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 396   = wceq 1539  wcel 2106  wne 2943  wral 3064  {crab 3068  cun 3885  wss 3887  c0 4256  𝒫 cpw 4533  {csn 4561   cuni 4839   cint 4879   class class class wbr 5074  cima 5592  Ord word 6265  suc csuc 6268   Fn wfn 6428  cfv 6433  (class class class)co 7275  1oc1o 8290   No csur 33843   <s cslt 33844   bday cbday 33845   <<s csslt 33975   |s cscut 33977   0s c0s 34016   1s c1s 34017
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1o 8297  df-2o 8298  df-no 33846  df-slt 33847  df-bday 33848  df-sslt 33976  df-scut 33978  df-0s 34018  df-1s 34019
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator