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Theorem n0sbday 28355
Description: A non-negative surreal integer has a finite birthday. (Contributed by Scott Fenton, 18-Apr-2025.)
Assertion
Ref Expression
n0sbday (𝐴 ∈ ℕ0s → ( bday 𝐴) ∈ ω)

Proof of Theorem n0sbday
Dummy variables 𝑛 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6905 . . 3 (𝑚 = 0s → ( bday 𝑚) = ( bday ‘ 0s ))
21eleq1d 2825 . 2 (𝑚 = 0s → (( bday 𝑚) ∈ ω ↔ ( bday ‘ 0s ) ∈ ω))
3 fveq2 6905 . . 3 (𝑚 = 𝑛 → ( bday 𝑚) = ( bday 𝑛))
43eleq1d 2825 . 2 (𝑚 = 𝑛 → (( bday 𝑚) ∈ ω ↔ ( bday 𝑛) ∈ ω))
5 fveq2 6905 . . 3 (𝑚 = (𝑛 +s 1s ) → ( bday 𝑚) = ( bday ‘(𝑛 +s 1s )))
65eleq1d 2825 . 2 (𝑚 = (𝑛 +s 1s ) → (( bday 𝑚) ∈ ω ↔ ( bday ‘(𝑛 +s 1s )) ∈ ω))
7 fveq2 6905 . . 3 (𝑚 = 𝐴 → ( bday 𝑚) = ( bday 𝐴))
87eleq1d 2825 . 2 (𝑚 = 𝐴 → (( bday 𝑚) ∈ ω ↔ ( bday 𝐴) ∈ ω))
9 bday0s 27874 . . 3 ( bday ‘ 0s ) = ∅
10 peano1 7911 . . 3 ∅ ∈ ω
119, 10eqeltri 2836 . 2 ( bday ‘ 0s ) ∈ ω
12 peano2n0s 28336 . . . . . . . . 9 (𝑛 ∈ ℕ0s → (𝑛 +s 1s ) ∈ ℕ0s)
13 n0scut 28339 . . . . . . . . 9 ((𝑛 +s 1s ) ∈ ℕ0s → (𝑛 +s 1s ) = ({((𝑛 +s 1s ) -s 1s )} |s ∅))
1412, 13syl 17 . . . . . . . 8 (𝑛 ∈ ℕ0s → (𝑛 +s 1s ) = ({((𝑛 +s 1s ) -s 1s )} |s ∅))
15 n0sno 28329 . . . . . . . . . . 11 (𝑛 ∈ ℕ0s𝑛 No )
16 1sno 27873 . . . . . . . . . . 11 1s No
17 pncans 28103 . . . . . . . . . . 11 ((𝑛 No ∧ 1s No ) → ((𝑛 +s 1s ) -s 1s ) = 𝑛)
1815, 16, 17sylancl 586 . . . . . . . . . 10 (𝑛 ∈ ℕ0s → ((𝑛 +s 1s ) -s 1s ) = 𝑛)
1918sneqd 4637 . . . . . . . . 9 (𝑛 ∈ ℕ0s → {((𝑛 +s 1s ) -s 1s )} = {𝑛})
2019oveq1d 7447 . . . . . . . 8 (𝑛 ∈ ℕ0s → ({((𝑛 +s 1s ) -s 1s )} |s ∅) = ({𝑛} |s ∅))
2114, 20eqtrd 2776 . . . . . . 7 (𝑛 ∈ ℕ0s → (𝑛 +s 1s ) = ({𝑛} |s ∅))
2221fveq2d 6909 . . . . . 6 (𝑛 ∈ ℕ0s → ( bday ‘(𝑛 +s 1s )) = ( bday ‘({𝑛} |s ∅)))
23 snelpwi 5447 . . . . . . . 8 (𝑛 No → {𝑛} ∈ 𝒫 No )
24 nulssgt 27844 . . . . . . . 8 ({𝑛} ∈ 𝒫 No → {𝑛} <<s ∅)
2515, 23, 243syl 18 . . . . . . 7 (𝑛 ∈ ℕ0s → {𝑛} <<s ∅)
26 un0 4393 . . . . . . . . . 10 ({𝑛} ∪ ∅) = {𝑛}
2726imaeq2i 6075 . . . . . . . . 9 ( bday “ ({𝑛} ∪ ∅)) = ( bday “ {𝑛})
28 bdayfn 27819 . . . . . . . . . 10 bday Fn No
29 fnsnfv 6987 . . . . . . . . . 10 (( bday Fn No 𝑛 No ) → {( bday 𝑛)} = ( bday “ {𝑛}))
3028, 15, 29sylancr 587 . . . . . . . . 9 (𝑛 ∈ ℕ0s → {( bday 𝑛)} = ( bday “ {𝑛}))
3127, 30eqtr4id 2795 . . . . . . . 8 (𝑛 ∈ ℕ0s → ( bday “ ({𝑛} ∪ ∅)) = {( bday 𝑛)})
32 fvex 6918 . . . . . . . . . 10 ( bday 𝑛) ∈ V
3332sucid 6465 . . . . . . . . 9 ( bday 𝑛) ∈ suc ( bday 𝑛)
34 snssi 4807 . . . . . . . . 9 (( bday 𝑛) ∈ suc ( bday 𝑛) → {( bday 𝑛)} ⊆ suc ( bday 𝑛))
3533, 34ax-mp 5 . . . . . . . 8 {( bday 𝑛)} ⊆ suc ( bday 𝑛)
3631, 35eqsstrdi 4027 . . . . . . 7 (𝑛 ∈ ℕ0s → ( bday “ ({𝑛} ∪ ∅)) ⊆ suc ( bday 𝑛))
37 bdayelon 27822 . . . . . . . . 9 ( bday 𝑛) ∈ On
3837onsuci 7860 . . . . . . . 8 suc ( bday 𝑛) ∈ On
39 scutbdaybnd 27861 . . . . . . . 8 (({𝑛} <<s ∅ ∧ suc ( bday 𝑛) ∈ On ∧ ( bday “ ({𝑛} ∪ ∅)) ⊆ suc ( bday 𝑛)) → ( bday ‘({𝑛} |s ∅)) ⊆ suc ( bday 𝑛))
4038, 39mp3an2 1450 . . . . . . 7 (({𝑛} <<s ∅ ∧ ( bday “ ({𝑛} ∪ ∅)) ⊆ suc ( bday 𝑛)) → ( bday ‘({𝑛} |s ∅)) ⊆ suc ( bday 𝑛))
4125, 36, 40syl2anc 584 . . . . . 6 (𝑛 ∈ ℕ0s → ( bday ‘({𝑛} |s ∅)) ⊆ suc ( bday 𝑛))
4222, 41eqsstrd 4017 . . . . 5 (𝑛 ∈ ℕ0s → ( bday ‘(𝑛 +s 1s )) ⊆ suc ( bday 𝑛))
43 bdayelon 27822 . . . . . 6 ( bday ‘(𝑛 +s 1s )) ∈ On
44 onsssuc 6473 . . . . . 6 ((( bday ‘(𝑛 +s 1s )) ∈ On ∧ suc ( bday 𝑛) ∈ On) → (( bday ‘(𝑛 +s 1s )) ⊆ suc ( bday 𝑛) ↔ ( bday ‘(𝑛 +s 1s )) ∈ suc suc ( bday 𝑛)))
4543, 38, 44mp2an 692 . . . . 5 (( bday ‘(𝑛 +s 1s )) ⊆ suc ( bday 𝑛) ↔ ( bday ‘(𝑛 +s 1s )) ∈ suc suc ( bday 𝑛))
4642, 45sylib 218 . . . 4 (𝑛 ∈ ℕ0s → ( bday ‘(𝑛 +s 1s )) ∈ suc suc ( bday 𝑛))
47 peano2 7913 . . . . 5 (( bday 𝑛) ∈ ω → suc ( bday 𝑛) ∈ ω)
48 peano2 7913 . . . . 5 (suc ( bday 𝑛) ∈ ω → suc suc ( bday 𝑛) ∈ ω)
4947, 48syl 17 . . . 4 (( bday 𝑛) ∈ ω → suc suc ( bday 𝑛) ∈ ω)
50 elnn 7899 . . . 4 ((( bday ‘(𝑛 +s 1s )) ∈ suc suc ( bday 𝑛) ∧ suc suc ( bday 𝑛) ∈ ω) → ( bday ‘(𝑛 +s 1s )) ∈ ω)
5146, 49, 50syl2an 596 . . 3 ((𝑛 ∈ ℕ0s ∧ ( bday 𝑛) ∈ ω) → ( bday ‘(𝑛 +s 1s )) ∈ ω)
5251ex 412 . 2 (𝑛 ∈ ℕ0s → (( bday 𝑛) ∈ ω → ( bday ‘(𝑛 +s 1s )) ∈ ω))
532, 4, 6, 8, 11, 52n0sind 28338 1 (𝐴 ∈ ℕ0s → ( bday 𝐴) ∈ ω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1539  wcel 2107  cun 3948  wss 3950  c0 4332  𝒫 cpw 4599  {csn 4625   class class class wbr 5142  cima 5687  Oncon0 6383  suc csuc 6385   Fn wfn 6555  cfv 6560  (class class class)co 7432  ωcom 7888   No csur 27685   bday cbday 27687   <<s csslt 27826   |s cscut 27828   0s c0s 27868   1s c1s 27869   +s cadds 27993   -s csubs 28053  0scnn0s 28319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-ot 4634  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-nadd 8705  df-no 27688  df-slt 27689  df-bday 27690  df-sle 27791  df-sslt 27827  df-scut 27829  df-0s 27870  df-1s 27871  df-made 27887  df-old 27888  df-left 27890  df-right 27891  df-norec 27972  df-norec2 27983  df-adds 27994  df-negs 28054  df-subs 28055  df-n0s 28321
This theorem is referenced by:  n0ssold  28356  zsbday  28393
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