MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  newbday Structured version   Visualization version   GIF version

Theorem newbday 27940
Description: A surreal is an element of a new set iff its birthday is equal to that ordinal. Remark in [Conway] p. 29. (Contributed by Scott Fenton, 19-Aug-2024.)
Assertion
Ref Expression
newbday ((𝐴 ∈ On ∧ 𝑋 No ) → (𝑋 ∈ ( N ‘𝐴) ↔ ( bday 𝑋) = 𝐴))

Proof of Theorem newbday
StepHypRef Expression
1 madebday 27938 . . 3 ((𝐴 ∈ On ∧ 𝑋 No ) → (𝑋 ∈ ( M ‘𝐴) ↔ ( bday 𝑋) ⊆ 𝐴))
2 oldbday 27939 . . . 4 ((𝐴 ∈ On ∧ 𝑋 No ) → (𝑋 ∈ ( O ‘𝐴) ↔ ( bday 𝑋) ∈ 𝐴))
32notbid 318 . . 3 ((𝐴 ∈ On ∧ 𝑋 No ) → (¬ 𝑋 ∈ ( O ‘𝐴) ↔ ¬ ( bday 𝑋) ∈ 𝐴))
41, 3anbi12d 632 . 2 ((𝐴 ∈ On ∧ 𝑋 No ) → ((𝑋 ∈ ( M ‘𝐴) ∧ ¬ 𝑋 ∈ ( O ‘𝐴)) ↔ (( bday 𝑋) ⊆ 𝐴 ∧ ¬ ( bday 𝑋) ∈ 𝐴)))
5 newval 27894 . . . . . 6 ( N ‘𝐴) = (( M ‘𝐴) ∖ ( O ‘𝐴))
65a1i 11 . . . . 5 (𝐴 ∈ On → ( N ‘𝐴) = (( M ‘𝐴) ∖ ( O ‘𝐴)))
76eleq2d 2827 . . . 4 (𝐴 ∈ On → (𝑋 ∈ ( N ‘𝐴) ↔ 𝑋 ∈ (( M ‘𝐴) ∖ ( O ‘𝐴))))
8 eldif 3961 . . . 4 (𝑋 ∈ (( M ‘𝐴) ∖ ( O ‘𝐴)) ↔ (𝑋 ∈ ( M ‘𝐴) ∧ ¬ 𝑋 ∈ ( O ‘𝐴)))
97, 8bitrdi 287 . . 3 (𝐴 ∈ On → (𝑋 ∈ ( N ‘𝐴) ↔ (𝑋 ∈ ( M ‘𝐴) ∧ ¬ 𝑋 ∈ ( O ‘𝐴))))
109adantr 480 . 2 ((𝐴 ∈ On ∧ 𝑋 No ) → (𝑋 ∈ ( N ‘𝐴) ↔ (𝑋 ∈ ( M ‘𝐴) ∧ ¬ 𝑋 ∈ ( O ‘𝐴))))
11 bdayelon 27821 . . . . 5 ( bday 𝑋) ∈ On
1211onordi 6495 . . . 4 Ord ( bday 𝑋)
13 eloni 6394 . . . 4 (𝐴 ∈ On → Ord 𝐴)
14 ordtri4 6421 . . . 4 ((Ord ( bday 𝑋) ∧ Ord 𝐴) → (( bday 𝑋) = 𝐴 ↔ (( bday 𝑋) ⊆ 𝐴 ∧ ¬ ( bday 𝑋) ∈ 𝐴)))
1512, 13, 14sylancr 587 . . 3 (𝐴 ∈ On → (( bday 𝑋) = 𝐴 ↔ (( bday 𝑋) ⊆ 𝐴 ∧ ¬ ( bday 𝑋) ∈ 𝐴)))
1615adantr 480 . 2 ((𝐴 ∈ On ∧ 𝑋 No ) → (( bday 𝑋) = 𝐴 ↔ (( bday 𝑋) ⊆ 𝐴 ∧ ¬ ( bday 𝑋) ∈ 𝐴)))
174, 10, 163bitr4d 311 1 ((𝐴 ∈ On ∧ 𝑋 No ) → (𝑋 ∈ ( N ‘𝐴) ↔ ( bday 𝑋) = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  cdif 3948  wss 3951  Ord word 6383  Oncon0 6384  cfv 6561   No csur 27684   bday cbday 27686   M cmade 27881   O cold 27882   N cnew 27883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-1o 8506  df-2o 8507  df-no 27687  df-slt 27688  df-bday 27689  df-sslt 27826  df-scut 27828  df-made 27886  df-old 27887  df-new 27888  df-left 27889  df-right 27890
This theorem is referenced by:  sltonold  28283
  Copyright terms: Public domain W3C validator