Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  onnobdayg Structured version   Visualization version   GIF version

Theorem onnobdayg 43421
Description: Every ordinal maps to a surreal number of that birthday. (Contributed by RP, 21-Sep-2023.)
Assertion
Ref Expression
onnobdayg ((𝐴 ∈ On ∧ 𝐵 ∈ {1o, 2o}) → ( bday ‘(𝐴 × {𝐵})) = 𝐴)

Proof of Theorem onnobdayg
StepHypRef Expression
1 onnog 43420 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ {1o, 2o}) → (𝐴 × {𝐵}) ∈ No )
2 bdayval 27683 . . 3 ((𝐴 × {𝐵}) ∈ No → ( bday ‘(𝐴 × {𝐵})) = dom (𝐴 × {𝐵}))
31, 2syl 17 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ {1o, 2o}) → ( bday ‘(𝐴 × {𝐵})) = dom (𝐴 × {𝐵}))
4 simpr 484 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ {1o, 2o}) → 𝐵 ∈ {1o, 2o})
5 snnzg 4772 . . 3 (𝐵 ∈ {1o, 2o} → {𝐵} ≠ ∅)
6 dmxp 5937 . . 3 ({𝐵} ≠ ∅ → dom (𝐴 × {𝐵}) = 𝐴)
74, 5, 63syl 18 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ {1o, 2o}) → dom (𝐴 × {𝐵}) = 𝐴)
83, 7eqtrd 2776 1 ((𝐴 ∈ On ∧ 𝐵 ∈ {1o, 2o}) → ( bday ‘(𝐴 × {𝐵})) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wne 2939  c0 4332  {csn 4624  {cpr 4626   × cxp 5681  dom cdm 5683  Oncon0 6382  cfv 6559  1oc1o 8495  2oc2o 8496   No csur 27674   bday cbday 27676
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 2707  ax-sep 5294  ax-nul 5304  ax-pow 5363  ax-pr 5430  ax-un 7751
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  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-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-br 5142  df-opab 5204  df-mpt 5224  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-iota 6512  df-fun 6561  df-fn 6562  df-f 6563  df-fv 6567  df-no 27677  df-bday 27679
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator