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Theorem bday1s 27557
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 27551 . . 3 1s = ({ 0s } |s ∅)
21fveq2i 6894 . 2 ( bday ‘ 1s ) = ( bday ‘({ 0s } |s ∅))
3 0sno 27552 . . . . . . 7 0s No
4 snelpwi 5443 . . . . . . 7 ( 0s No → { 0s } ∈ 𝒫 No )
53, 4ax-mp 5 . . . . . 6 { 0s } ∈ 𝒫 No
6 nulssgt 27524 . . . . . 6 ({ 0s } ∈ 𝒫 No → { 0s } <<s ∅)
75, 6ax-mp 5 . . . . 5 { 0s } <<s ∅
8 scutbdaybnd2 27542 . . . . 5 ({ 0s } <<s ∅ → ( bday ‘({ 0s } |s ∅)) ⊆ suc ( bday “ ({ 0s } ∪ ∅)))
97, 8ax-mp 5 . . . 4 ( bday ‘({ 0s } |s ∅)) ⊆ suc ( bday “ ({ 0s } ∪ ∅))
10 un0 4390 . . . . . . . . . 10 ({ 0s } ∪ ∅) = { 0s }
1110imaeq2i 6057 . . . . . . . . 9 ( bday “ ({ 0s } ∪ ∅)) = ( bday “ { 0s })
12 bdayfn 27499 . . . . . . . . . 10 bday Fn No
13 fnsnfv 6970 . . . . . . . . . 10 (( bday Fn No ∧ 0s No ) → {( bday ‘ 0s )} = ( bday “ { 0s }))
1412, 3, 13mp2an 690 . . . . . . . . 9 {( bday ‘ 0s )} = ( bday “ { 0s })
15 bday0s 27554 . . . . . . . . . 10 ( bday ‘ 0s ) = ∅
1615sneqi 4639 . . . . . . . . 9 {( bday ‘ 0s )} = {∅}
1711, 14, 163eqtr2i 2766 . . . . . . . 8 ( bday “ ({ 0s } ∪ ∅)) = {∅}
1817unieqi 4921 . . . . . . 7 ( bday “ ({ 0s } ∪ ∅)) = {∅}
19 0ex 5307 . . . . . . . 8 ∅ ∈ V
2019unisn 4930 . . . . . . 7 {∅} = ∅
2118, 20eqtri 2760 . . . . . 6 ( bday “ ({ 0s } ∪ ∅)) = ∅
22 suceq 6430 . . . . . 6 ( ( bday “ ({ 0s } ∪ ∅)) = ∅ → suc ( bday “ ({ 0s } ∪ ∅)) = suc ∅)
2321, 22ax-mp 5 . . . . 5 suc ( bday “ ({ 0s } ∪ ∅)) = suc ∅
24 df-1o 8468 . . . . 5 1o = suc ∅
2523, 24eqtr4i 2763 . . . 4 suc ( bday “ ({ 0s } ∪ ∅)) = 1o
269, 25sseqtri 4018 . . 3 ( bday ‘({ 0s } |s ∅)) ⊆ 1o
27 ssrab2 4077 . . . . . 6 {𝑥 No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)} ⊆ No
28 fnssintima 7361 . . . . . 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 4638 . . . . . . . . 9 (𝑥 = 𝑦 → {𝑥} = {𝑦})
3130breq2d 5160 . . . . . . . 8 (𝑥 = 𝑦 → ({ 0s } <<s {𝑥} ↔ { 0s } <<s {𝑦}))
3230breq1d 5158 . . . . . . . 8 (𝑥 = 𝑦 → ({𝑥} <<s ∅ ↔ {𝑦} <<s ∅))
3331, 32anbi12d 631 . . . . . . 7 (𝑥 = 𝑦 → (({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅) ↔ ({ 0s } <<s {𝑦} ∧ {𝑦} <<s ∅)))
3433elrab 3683 . . . . . 6 (𝑦 ∈ {𝑥 No ∣ ({ 0s } <<s {𝑥} ∧ {𝑥} <<s ∅)} ↔ (𝑦 No ∧ ({ 0s } <<s {𝑦} ∧ {𝑦} <<s ∅)))
35 sltirr 27473 . . . . . . . . . . . . 13 ( 0s No → ¬ 0s <s 0s )
363, 35ax-mp 5 . . . . . . . . . . . 12 ¬ 0s <s 0s
37 breq2 5152 . . . . . . . . . . . 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 27556 . . . . . . . . . . 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 27502 . . . . . . . . . . 11 ( bday 𝑦) ∈ On
4443onordi 6475 . . . . . . . . . 10 Ord ( bday 𝑦)
45 ordge1n0 8496 . . . . . . . . . 10 (Ord ( bday 𝑦) → (1o ⊆ ( bday 𝑦) ↔ ( bday 𝑦) ≠ ∅))
4644, 45ax-mp 5 . . . . . . . . 9 (1o ⊆ ( bday 𝑦) ↔ ( bday 𝑦) ≠ ∅)
4742, 46imbitrrdi 251 . . . . . . . 8 (𝑦 No → ( 0s <s 𝑦 → 1o ⊆ ( bday 𝑦)))
48 ssltsep 27516 . . . . . . . . 9 ({ 0s } <<s {𝑦} → ∀𝑥 ∈ { 0s }∀𝑧 ∈ {𝑦}𝑥 <s 𝑧)
49 vex 3478 . . . . . . . . . . . 12 𝑦 ∈ V
50 breq2 5152 . . . . . . . . . . . 12 (𝑧 = 𝑦 → (𝑥 <s 𝑧𝑥 <s 𝑦))
5149, 50ralsn 4685 . . . . . . . . . . 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 5151 . . . . . . . . . . 11 (𝑥 = 0s → (𝑥 <s 𝑦 ↔ 0s <s 𝑦))
5553, 54ralsn 4685 . . . . . . . . . 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 27530 . . . . 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 4019 . . 3 1o ⊆ ( bday ‘({ 0s } |s ∅))
6526, 64eqssi 3998 . 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 3946  wss 3948  c0 4322  𝒫 cpw 4602  {csn 4628   cuni 4908   cint 4950   class class class wbr 5148  cima 5679  Ord word 6363  suc csuc 6366   Fn wfn 6538  cfv 6543  (class class class)co 7411  1oc1o 8461   No csur 27367   <s cslt 27368   bday cbday 27369   <<s csslt 27506   |s cscut 27508   0s c0s 27548   1s c1s 27549
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 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7727
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 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  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 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1o 8468  df-2o 8469  df-no 27370  df-slt 27371  df-bday 27372  df-sslt 27507  df-scut 27509  df-0s 27550  df-1s 27551
This theorem is referenced by:  cuteq1  27559  left1s  27614  right1s  27615
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