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Theorem bday0s 27189
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 27185 . . . 4 0s = (∅ |s ∅)
21fveq2i 6846 . . 3 ( bday ‘ 0s ) = ( bday ‘(∅ |s ∅))
3 0elpw 5312 . . . 4 ∅ ∈ 𝒫 No
4 nulssgt 27159 . . . 4 (∅ ∈ 𝒫 No → ∅ <<s ∅)
5 scutbday 27165 . . . 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 2761 . 2 ( bday ‘ 0s ) = ( bday “ {𝑥 No ∣ (∅ <<s {𝑥} ∧ {𝑥} <<s ∅)})
8 snelpwi 5401 . . . . . . . 8 (𝑥 No → {𝑥} ∈ 𝒫 No )
9 nulsslt 27158 . . . . . . . . 9 ({𝑥} ∈ 𝒫 No → ∅ <<s {𝑥})
10 nulssgt 27159 . . . . . . . . 9 ({𝑥} ∈ 𝒫 No → {𝑥} <<s ∅)
119, 10jca 513 . . . . . . . 8 ({𝑥} ∈ 𝒫 No → (∅ <<s {𝑥} ∧ {𝑥} <<s ∅))
128, 11syl 17 . . . . . . 7 (𝑥 No → (∅ <<s {𝑥} ∧ {𝑥} <<s ∅))
1312rabeqc 3418 . . . . . 6 {𝑥 No ∣ (∅ <<s {𝑥} ∧ {𝑥} <<s ∅)} = No
14 bdaydm 27136 . . . . . 6 dom bday = No
1513, 14eqtr4i 2764 . . . . 5 {𝑥 No ∣ (∅ <<s {𝑥} ∧ {𝑥} <<s ∅)} = dom bday
1615imaeq2i 6012 . . . 4 ( bday “ {𝑥 No ∣ (∅ <<s {𝑥} ∧ {𝑥} <<s ∅)}) = ( bday “ dom bday )
17 imadmrn 6024 . . . 4 ( bday “ dom bday ) = ran bday
18 bdayrn 27137 . . . 4 ran bday = On
1916, 17, 183eqtri 2765 . . 3 ( bday “ {𝑥 No ∣ (∅ <<s {𝑥} ∧ {𝑥} <<s ∅)}) = On
2019inteqi 4912 . 2 ( bday “ {𝑥 No ∣ (∅ <<s {𝑥} ∧ {𝑥} <<s ∅)}) = On
21 inton 6376 . 2 On = ∅
227, 20, 213eqtri 2765 1 ( bday ‘ 0s ) = ∅
Colors of variables: wff setvar class
Syntax hints:  wa 397   = wceq 1542  wcel 2107  {crab 3406  c0 4283  𝒫 cpw 4561  {csn 4587   cint 4908   class class class wbr 5106  dom cdm 5634  ran crn 5635  cima 5637  Oncon0 6318  cfv 6497  (class class class)co 7358   No csur 27004   bday cbday 27006   <<s csslt 27142   |s cscut 27144   0s c0s 27183
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
This theorem is referenced by:  bday0b  27191  bday1s  27192  cuteq0  27193  left0s  27244  right0s  27245  addsproplem2  27304  negsproplem2  27349  negsproplem6  27353  mulsproplem2  27402  mulsproplem3  27403  mulsproplem4  27404  mulsproplem5  27405
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