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Theorem bday1s 27192
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 27186 . . 3 1s = ({ 0s } |s ∅)
21fveq2i 6846 . 2 ( bday ‘ 1s ) = ( bday ‘({ 0s } |s ∅))
3 0sno 27187 . . . . . . 7 0s No
4 snelpwi 5401 . . . . . . 7 ( 0s No → { 0s } ∈ 𝒫 No )
53, 4ax-mp 5 . . . . . 6 { 0s } ∈ 𝒫 No
6 nulssgt 27159 . . . . . 6 ({ 0s } ∈ 𝒫 No → { 0s } <<s ∅)
75, 6ax-mp 5 . . . . 5 { 0s } <<s ∅
8 scutbdaybnd2 27177 . . . . 5 ({ 0s } <<s ∅ → ( bday ‘({ 0s } |s ∅)) ⊆ suc ( bday “ ({ 0s } ∪ ∅)))
97, 8ax-mp 5 . . . 4 ( bday ‘({ 0s } |s ∅)) ⊆ suc ( bday “ ({ 0s } ∪ ∅))
10 un0 4351 . . . . . . . . . 10 ({ 0s } ∪ ∅) = { 0s }
1110imaeq2i 6012 . . . . . . . . 9 ( bday “ ({ 0s } ∪ ∅)) = ( bday “ { 0s })
12 bdayfn 27135 . . . . . . . . . 10 bday Fn No
13 fnsnfv 6921 . . . . . . . . . 10 (( bday Fn No ∧ 0s No ) → {( bday ‘ 0s )} = ( bday “ { 0s }))
1412, 3, 13mp2an 691 . . . . . . . . 9 {( bday ‘ 0s )} = ( bday “ { 0s })
15 bday0s 27189 . . . . . . . . . 10 ( bday ‘ 0s ) = ∅
1615sneqi 4598 . . . . . . . . 9 {( bday ‘ 0s )} = {∅}
1711, 14, 163eqtr2i 2767 . . . . . . . 8 ( bday “ ({ 0s } ∪ ∅)) = {∅}
1817unieqi 4879 . . . . . . 7 ( bday “ ({ 0s } ∪ ∅)) = {∅}
19 0ex 5265 . . . . . . . 8 ∅ ∈ V
2019unisn 4888 . . . . . . 7 {∅} = ∅
2118, 20eqtri 2761 . . . . . 6 ( bday “ ({ 0s } ∪ ∅)) = ∅
22 suceq 6384 . . . . . 6 ( ( bday “ ({ 0s } ∪ ∅)) = ∅ → suc ( bday “ ({ 0s } ∪ ∅)) = suc ∅)
2321, 22ax-mp 5 . . . . 5 suc ( bday “ ({ 0s } ∪ ∅)) = suc ∅
24 df-1o 8413 . . . . 5 1o = suc ∅
2523, 24eqtr4i 2764 . . . 4 suc ( bday “ ({ 0s } ∪ ∅)) = 1o
269, 25sseqtri 3981 . . 3 ( bday ‘({ 0s } |s ∅)) ⊆ 1o
27 ssrab2 4038 . . . . . 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)))
2912, 27, 28mp2an 691 . . . . 5 (1o ( bday “ {𝑥 No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)}) ↔ ∀𝑦 ∈ {𝑥 No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)}1o ⊆ ( bday 𝑦))
30 sneq 4597 . . . . . . . . 9 (𝑥 = 𝑦 → {𝑥} = {𝑦})
3130breq2d 5118 . . . . . . . 8 (𝑥 = 𝑦 → ({ 0s } <<s {𝑥} ↔ { 0s } <<s {𝑦}))
3230breq1d 5116 . . . . . . . 8 (𝑥 = 𝑦 → ({𝑥} <<s ∅ ↔ {𝑦} <<s ∅))
3331, 32anbi12d 632 . . . . . . 7 (𝑥 = 𝑦 → (({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅) ↔ ({ 0s } <<s {𝑦} ∧ {𝑦} <<s ∅)))
3433elrab 3646 . . . . . 6 (𝑦 ∈ {𝑥 No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)} ↔ (𝑦 No ∧ ({ 0s } <<s {𝑦} ∧ {𝑦} <<s ∅)))
35 sltirr 27110 . . . . . . . . . . . . 13 ( 0s No → ¬ 0s <s 0s )
363, 35ax-mp 5 . . . . . . . . . . . 12 ¬ 0s <s 0s
37 breq2 5110 . . . . . . . . . . . 12 (𝑦 = 0s → ( 0s <s 𝑦 ↔ 0s <s 0s ))
3836, 37mtbiri 327 . . . . . . . . . . 11 (𝑦 = 0s → ¬ 0s <s 𝑦)
3938necon2ai 2970 . . . . . . . . . 10 ( 0s <s 𝑦𝑦 ≠ 0s )
40 bday0b 27191 . . . . . . . . . . 11 (𝑦 No → (( bday 𝑦) = ∅ ↔ 𝑦 = 0s ))
4140necon3bid 2985 . . . . . . . . . 10 (𝑦 No → (( bday 𝑦) ≠ ∅ ↔ 𝑦 ≠ 0s ))
4239, 41syl5ibr 246 . . . . . . . . 9 (𝑦 No → ( 0s <s 𝑦 → ( bday 𝑦) ≠ ∅))
43 bdayelon 27138 . . . . . . . . . . 11 ( bday 𝑦) ∈ On
4443onordi 6429 . . . . . . . . . 10 Ord ( bday 𝑦)
45 ordge1n0 8441 . . . . . . . . . 10 (Ord ( bday 𝑦) → (1o ⊆ ( bday 𝑦) ↔ ( bday 𝑦) ≠ ∅))
4644, 45ax-mp 5 . . . . . . . . 9 (1o ⊆ ( bday 𝑦) ↔ ( bday 𝑦) ≠ ∅)
4742, 46syl6ibr 252 . . . . . . . 8 (𝑦 No → ( 0s <s 𝑦 → 1o ⊆ ( bday 𝑦)))
48 ssltsep 27152 . . . . . . . . 9 ({ 0s } <<s {𝑦} → ∀𝑥 ∈ { 0s }∀𝑧 ∈ {𝑦}𝑥 <s 𝑧)
49 vex 3448 . . . . . . . . . . . 12 𝑦 ∈ V
50 breq2 5110 . . . . . . . . . . . 12 (𝑧 = 𝑦 → (𝑥 <s 𝑧𝑥 <s 𝑦))
5149, 50ralsn 4643 . . . . . . . . . . 11 (∀𝑧 ∈ {𝑦}𝑥 <s 𝑧𝑥 <s 𝑦)
5251ralbii 3093 . . . . . . . . . 10 (∀𝑥 ∈ { 0s }∀𝑧 ∈ {𝑦}𝑥 <s 𝑧 ↔ ∀𝑥 ∈ { 0s }𝑥 <s 𝑦)
533elexi 3463 . . . . . . . . . . 11 0s ∈ V
54 breq1 5109 . . . . . . . . . . 11 (𝑥 = 0s → (𝑥 <s 𝑦 ↔ 0s <s 𝑦))
5553, 54ralsn 4643 . . . . . . . . . 10 (∀𝑥 ∈ { 0s }𝑥 <s 𝑦 ↔ 0s <s 𝑦)
5652, 55bitri 275 . . . . . . . . 9 (∀𝑥 ∈ { 0s }∀𝑧 ∈ {𝑦}𝑥 <s 𝑧 ↔ 0s <s 𝑦)
5748, 56sylib 217 . . . . . . . 8 ({ 0s } <<s {𝑦} → 0s <s 𝑦)
5847, 57impel 507 . . . . . . 7 ((𝑦 No ∧ { 0s } <<s {𝑦}) → 1o ⊆ ( bday 𝑦))
5958adantrr 716 . . . . . 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 3068 . . . 4 1o ( bday “ {𝑥 No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)})
62 scutbday 27165 . . . . 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 3982 . . 3 1o ⊆ ( bday ‘({ 0s } |s ∅))
6526, 64eqssi 3961 . 2 ( bday ‘({ 0s } |s ∅)) = 1o
662, 65eqtri 2761 1 ( bday ‘ 1s ) = 1o
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 397   = wceq 1542  wcel 2107  wne 2940  wral 3061  {crab 3406  cun 3909  wss 3911  c0 4283  𝒫 cpw 4561  {csn 4587   cuni 4866   cint 4908   class class class wbr 5106  cima 5637  Ord word 6317  suc csuc 6320   Fn wfn 6492  cfv 6497  (class class class)co 7358  1oc1o 8406   No csur 27004   <s cslt 27005   bday cbday 27006   <<s csslt 27142   |s cscut 27144   0s c0s 27183   1s c1s 27184
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ord 6321  df-on 6322  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1o 8413  df-2o 8414  df-no 27007  df-slt 27008  df-bday 27009  df-sslt 27143  df-scut 27145  df-0s 27185  df-1s 27186
This theorem is referenced by:  left1s  27246  right1s  27247
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