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

Theorem bdaybndex 43449
Description: Bounds formed from the birthday are surreal numbers. (Contributed by RP, 21-Sep-2023.)
Assertion
Ref Expression
bdaybndex ((𝐴 No 𝐵 = ( bday 𝐴) ∧ 𝐶 ∈ {1o, 2o}) → (𝐵 × {𝐶}) ∈ No )

Proof of Theorem bdaybndex
StepHypRef Expression
1 simpr 484 . . . 4 ((𝐴 No 𝐵 = ( bday 𝐴)) → 𝐵 = ( bday 𝐴))
2 bdayval 27694 . . . . 5 (𝐴 No → ( bday 𝐴) = dom 𝐴)
32adantr 480 . . . 4 ((𝐴 No 𝐵 = ( bday 𝐴)) → ( bday 𝐴) = dom 𝐴)
41, 3eqtrd 2776 . . 3 ((𝐴 No 𝐵 = ( bday 𝐴)) → 𝐵 = dom 𝐴)
5 nodmon 27696 . . . 4 (𝐴 No → dom 𝐴 ∈ On)
65adantr 480 . . 3 ((𝐴 No 𝐵 = ( bday 𝐴)) → dom 𝐴 ∈ On)
74, 6eqeltrd 2840 . 2 ((𝐴 No 𝐵 = ( bday 𝐴)) → 𝐵 ∈ On)
8 onnog 43447 . 2 ((𝐵 ∈ On ∧ 𝐶 ∈ {1o, 2o}) → (𝐵 × {𝐶}) ∈ No )
97, 8stoic3 1775 1 ((𝐴 No 𝐵 = ( bday 𝐴) ∧ 𝐶 ∈ {1o, 2o}) → (𝐵 × {𝐶}) ∈ No )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1539  wcel 2107  {csn 4625  {cpr 4627   × cxp 5682  dom cdm 5684  Oncon0 6383  cfv 6560  1oc1o 8500  2oc2o 8501   No csur 27685   bday cbday 27687
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-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-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-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 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fv 6568  df-no 27688  df-bday 27690
This theorem is referenced by:  bdaybndbday  43450
  Copyright terms: Public domain W3C validator