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Theorem bday0s 34022
Description: Calculate the birthday of surreal zero. (Contributed by Scott Fenton, 7-Aug-2024.)
Assertion
Ref Expression
bday0s ( bday ‘ 0s ) = ∅

Proof of Theorem bday0s
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-0s 34018 . . . 4 0s = (∅ |s ∅)
21fveq2i 6777 . . 3 ( bday ‘ 0s ) = ( bday ‘(∅ |s ∅))
3 0elpw 5278 . . . 4 ∅ ∈ 𝒫 No
4 nulssgt 33992 . . . 4 (∅ ∈ 𝒫 No → ∅ <<s ∅)
5 scutbday 33998 . . . 4 (∅ <<s ∅ → ( bday ‘(∅ |s ∅)) = ( bday “ {𝑥 No ∣ (∅ <<s {𝑥} ∧ {𝑥} <<s ∅)}))
63, 4, 5mp2b 10 . . 3 ( bday ‘(∅ |s ∅)) = ( bday “ {𝑥 No ∣ (∅ <<s {𝑥} ∧ {𝑥} <<s ∅)})
72, 6eqtri 2766 . 2 ( bday ‘ 0s ) = ( bday “ {𝑥 No ∣ (∅ <<s {𝑥} ∧ {𝑥} <<s ∅)})
8 snelpwi 5360 . . . . . . . 8 (𝑥 No → {𝑥} ∈ 𝒫 No )
9 nulsslt 33991 . . . . . . . . 9 ({𝑥} ∈ 𝒫 No → ∅ <<s {𝑥})
10 nulssgt 33992 . . . . . . . . 9 ({𝑥} ∈ 𝒫 No → {𝑥} <<s ∅)
119, 10jca 512 . . . . . . . 8 ({𝑥} ∈ 𝒫 No → (∅ <<s {𝑥} ∧ {𝑥} <<s ∅))
128, 11syl 17 . . . . . . 7 (𝑥 No → (∅ <<s {𝑥} ∧ {𝑥} <<s ∅))
1312rabeqc 3622 . . . . . 6 {𝑥 No ∣ (∅ <<s {𝑥} ∧ {𝑥} <<s ∅)} = No
14 bdaydm 33969 . . . . . 6 dom bday = No
1513, 14eqtr4i 2769 . . . . 5 {𝑥 No ∣ (∅ <<s {𝑥} ∧ {𝑥} <<s ∅)} = dom bday
1615imaeq2i 5967 . . . 4 ( bday “ {𝑥 No ∣ (∅ <<s {𝑥} ∧ {𝑥} <<s ∅)}) = ( bday “ dom bday )
17 imadmrn 5979 . . . 4 ( bday “ dom bday ) = ran bday
18 bdayrn 33970 . . . 4 ran bday = On
1916, 17, 183eqtri 2770 . . 3 ( bday “ {𝑥 No ∣ (∅ <<s {𝑥} ∧ {𝑥} <<s ∅)}) = On
2019inteqi 4883 . 2 ( bday “ {𝑥 No ∣ (∅ <<s {𝑥} ∧ {𝑥} <<s ∅)}) = On
21 inton 6323 . 2 On = ∅
227, 20, 213eqtri 2770 1 ( bday ‘ 0s ) = ∅
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1539  wcel 2106  {crab 3068  c0 4256  𝒫 cpw 4533  {csn 4561   cint 4879   class class class wbr 5074  dom cdm 5589  ran crn 5590  cima 5592  Oncon0 6266  cfv 6433  (class class class)co 7275   No csur 33843   bday cbday 33845   <<s csslt 33975   |s cscut 33977   0s c0s 34016
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
This theorem is referenced by:  bday0b  34024  bday1s  34025  left0s  34075  right0s  34076
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