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Theorem bday1s 27261
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 27255 . . 3 1s = ({ 0s } |s ∅)
21fveq2i 6882 . 2 ( bday ‘ 1s ) = ( bday ‘({ 0s } |s ∅))
3 0sno 27256 . . . . . . 7 0s No
4 snelpwi 5437 . . . . . . 7 ( 0s No → { 0s } ∈ 𝒫 No )
53, 4ax-mp 5 . . . . . 6 { 0s } ∈ 𝒫 No
6 nulssgt 27228 . . . . . 6 ({ 0s } ∈ 𝒫 No → { 0s } <<s ∅)
75, 6ax-mp 5 . . . . 5 { 0s } <<s ∅
8 scutbdaybnd2 27246 . . . . 5 ({ 0s } <<s ∅ → ( bday ‘({ 0s } |s ∅)) ⊆ suc ( bday “ ({ 0s } ∪ ∅)))
97, 8ax-mp 5 . . . 4 ( bday ‘({ 0s } |s ∅)) ⊆ suc ( bday “ ({ 0s } ∪ ∅))
10 un0 4387 . . . . . . . . . 10 ({ 0s } ∪ ∅) = { 0s }
1110imaeq2i 6048 . . . . . . . . 9 ( bday “ ({ 0s } ∪ ∅)) = ( bday “ { 0s })
12 bdayfn 27204 . . . . . . . . . 10 bday Fn No
13 fnsnfv 6957 . . . . . . . . . 10 (( bday Fn No ∧ 0s No ) → {( bday ‘ 0s )} = ( bday “ { 0s }))
1412, 3, 13mp2an 690 . . . . . . . . 9 {( bday ‘ 0s )} = ( bday “ { 0s })
15 bday0s 27258 . . . . . . . . . 10 ( bday ‘ 0s ) = ∅
1615sneqi 4634 . . . . . . . . 9 {( bday ‘ 0s )} = {∅}
1711, 14, 163eqtr2i 2766 . . . . . . . 8 ( bday “ ({ 0s } ∪ ∅)) = {∅}
1817unieqi 4915 . . . . . . 7 ( bday “ ({ 0s } ∪ ∅)) = {∅}
19 0ex 5301 . . . . . . . 8 ∅ ∈ V
2019unisn 4924 . . . . . . 7 {∅} = ∅
2118, 20eqtri 2760 . . . . . 6 ( bday “ ({ 0s } ∪ ∅)) = ∅
22 suceq 6420 . . . . . 6 ( ( bday “ ({ 0s } ∪ ∅)) = ∅ → suc ( bday “ ({ 0s } ∪ ∅)) = suc ∅)
2321, 22ax-mp 5 . . . . 5 suc ( bday “ ({ 0s } ∪ ∅)) = suc ∅
24 df-1o 8450 . . . . 5 1o = suc ∅
2523, 24eqtr4i 2763 . . . 4 suc ( bday “ ({ 0s } ∪ ∅)) = 1o
269, 25sseqtri 4015 . . 3 ( bday ‘({ 0s } |s ∅)) ⊆ 1o
27 ssrab2 4074 . . . . . 6 {𝑥 No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)} ⊆ No
28 fnssintima 7344 . . . . . 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 690 . . . . 5 (1o ( bday “ {𝑥 No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)}) ↔ ∀𝑦 ∈ {𝑥 No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)}1o ⊆ ( bday 𝑦))
30 sneq 4633 . . . . . . . . 9 (𝑥 = 𝑦 → {𝑥} = {𝑦})
3130breq2d 5154 . . . . . . . 8 (𝑥 = 𝑦 → ({ 0s } <<s {𝑥} ↔ { 0s } <<s {𝑦}))
3230breq1d 5152 . . . . . . . 8 (𝑥 = 𝑦 → ({𝑥} <<s ∅ ↔ {𝑦} <<s ∅))
3331, 32anbi12d 631 . . . . . . 7 (𝑥 = 𝑦 → (({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅) ↔ ({ 0s } <<s {𝑦} ∧ {𝑦} <<s ∅)))
3433elrab 3680 . . . . . 6 (𝑦 ∈ {𝑥 No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)} ↔ (𝑦 No ∧ ({ 0s } <<s {𝑦} ∧ {𝑦} <<s ∅)))
35 sltirr 27178 . . . . . . . . . . . . 13 ( 0s No → ¬ 0s <s 0s )
363, 35ax-mp 5 . . . . . . . . . . . 12 ¬ 0s <s 0s
37 breq2 5146 . . . . . . . . . . . 12 (𝑦 = 0s → ( 0s <s 𝑦 ↔ 0s <s 0s ))
3836, 37mtbiri 326 . . . . . . . . . . 11 (𝑦 = 0s → ¬ 0s <s 𝑦)
3938necon2ai 2970 . . . . . . . . . 10 ( 0s <s 𝑦𝑦 ≠ 0s )
40 bday0b 27260 . . . . . . . . . . 11 (𝑦 No → (( bday 𝑦) = ∅ ↔ 𝑦 = 0s ))
4140necon3bid 2985 . . . . . . . . . 10 (𝑦 No → (( bday 𝑦) ≠ ∅ ↔ 𝑦 ≠ 0s ))
4239, 41imbitrrid 245 . . . . . . . . 9 (𝑦 No → ( 0s <s 𝑦 → ( bday 𝑦) ≠ ∅))
43 bdayelon 27207 . . . . . . . . . . 11 ( bday 𝑦) ∈ On
4443onordi 6465 . . . . . . . . . 10 Ord ( bday 𝑦)
45 ordge1n0 8478 . . . . . . . . . 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 27221 . . . . . . . . 9 ({ 0s } <<s {𝑦} → ∀𝑥 ∈ { 0s }∀𝑧 ∈ {𝑦}𝑥 <s 𝑧)
49 vex 3478 . . . . . . . . . . . 12 𝑦 ∈ V
50 breq2 5146 . . . . . . . . . . . 12 (𝑧 = 𝑦 → (𝑥 <s 𝑧𝑥 <s 𝑦))
5149, 50ralsn 4679 . . . . . . . . . . 11 (∀𝑧 ∈ {𝑦}𝑥 <s 𝑧𝑥 <s 𝑦)
5251ralbii 3093 . . . . . . . . . 10 (∀𝑥 ∈ { 0s }∀𝑧 ∈ {𝑦}𝑥 <s 𝑧 ↔ ∀𝑥 ∈ { 0s }𝑥 <s 𝑦)
533elexi 3493 . . . . . . . . . . 11 0s ∈ V
54 breq1 5145 . . . . . . . . . . 11 (𝑥 = 0s → (𝑥 <s 𝑦 ↔ 0s <s 𝑦))
5553, 54ralsn 4679 . . . . . . . . . 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 715 . . . . . 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 27234 . . . . 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 4016 . . 3 1o ⊆ ( bday ‘({ 0s } |s ∅))
6526, 64eqssi 3995 . 2 ( bday ‘({ 0s } |s ∅)) = 1o
662, 65eqtri 2760 1 ( bday ‘ 1s ) = 1o
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 396   = wceq 1541  wcel 2106  wne 2940  wral 3061  {crab 3432  cun 3943  wss 3945  c0 4319  𝒫 cpw 4597  {csn 4623   cuni 4902   cint 4944   class class class wbr 5142  cima 5673  Ord word 6353  suc csuc 6356   Fn wfn 6528  cfv 6533  (class class class)co 7394  1oc1o 8443   No csur 27072   <s cslt 27073   bday cbday 27074   <<s csslt 27211   |s cscut 27213   0s c0s 27252   1s c1s 27253
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pr 5421  ax-un 7709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4320  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5568  df-eprel 5574  df-po 5582  df-so 5583  df-fr 5625  df-we 5627  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-ord 6357  df-on 6358  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7350  df-ov 7397  df-oprab 7398  df-mpo 7399  df-1o 8450  df-2o 8451  df-no 27075  df-slt 27076  df-bday 27077  df-sslt 27212  df-scut 27214  df-0s 27254  df-1s 27255
This theorem is referenced by:  cuteq1  27263  left1s  27318  right1s  27319
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