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Theorem bday0s 33667
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 33663 . . . 4 0s = (∅ |s ∅)
21fveq2i 6679 . . 3 ( bday ‘ 0s ) = ( bday ‘(∅ |s ∅))
3 0elpw 5222 . . . 4 ∅ ∈ 𝒫 No
4 nulssgt 33637 . . . 4 (∅ ∈ 𝒫 No → ∅ <<s ∅)
5 scutbday 33643 . . . 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 5303 . . . . . . . 8 (𝑥 No → {𝑥} ∈ 𝒫 No )
9 nulsslt 33636 . . . . . . . . 9 ({𝑥} ∈ 𝒫 No → ∅ <<s {𝑥})
10 nulssgt 33637 . . . . . . . . 9 ({𝑥} ∈ 𝒫 No → {𝑥} <<s ∅)
119, 10jca 515 . . . . . . . 8 ({𝑥} ∈ 𝒫 No → (∅ <<s {𝑥} ∧ {𝑥} <<s ∅))
128, 11syl 17 . . . . . . 7 (𝑥 No → (∅ <<s {𝑥} ∧ {𝑥} <<s ∅))
1312rabeqc 3586 . . . . . 6 {𝑥 No ∣ (∅ <<s {𝑥} ∧ {𝑥} <<s ∅)} = No
14 bdaydm 33614 . . . . . 6 dom bday = No
1513, 14eqtr4i 2764 . . . . 5 {𝑥 No ∣ (∅ <<s {𝑥} ∧ {𝑥} <<s ∅)} = dom bday
1615imaeq2i 5901 . . . 4 ( bday “ {𝑥 No ∣ (∅ <<s {𝑥} ∧ {𝑥} <<s ∅)}) = ( bday “ dom bday )
17 imadmrn 5913 . . . 4 ( bday “ dom bday ) = ran bday
18 bdayrn 33615 . . . 4 ran bday = On
1916, 17, 183eqtri 2765 . . 3 ( bday “ {𝑥 No ∣ (∅ <<s {𝑥} ∧ {𝑥} <<s ∅)}) = On
2019inteqi 4840 . 2 ( bday “ {𝑥 No ∣ (∅ <<s {𝑥} ∧ {𝑥} <<s ∅)}) = On
21 inton 6229 . 2 On = ∅
227, 20, 213eqtri 2765 1 ( bday ‘ 0s ) = ∅
Colors of variables: wff setvar class
Syntax hints:  wa 399   = wceq 1542  wcel 2114  {crab 3057  c0 4211  𝒫 cpw 4488  {csn 4516   cint 4836   class class class wbr 5030  dom cdm 5525  ran crn 5526  cima 5528  Oncon0 6172  cfv 6339  (class class class)co 7172   No csur 33488   bday cbday 33490   <<s csslt 33620   |s cscut 33622   0s c0s 33661
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pr 5296  ax-un 7481
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-int 4837  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-ord 6175  df-on 6176  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7129  df-ov 7175  df-oprab 7176  df-mpo 7177  df-1o 8133  df-2o 8134  df-no 33491  df-slt 33492  df-bday 33493  df-sslt 33621  df-scut 33623  df-0s 33663
This theorem is referenced by:  bday0b  33669  bday1s  33670  left0s  33720  right0s  33721
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