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Theorem onno 43336
Description: Every ordinal maps to a surreal number. (Contributed by RP, 21-Sep-2023.)
Assertion
Ref Expression
onno (𝐴 ∈ On → (𝐴 × {2o}) ∈ No )

Proof of Theorem onno
StepHypRef Expression
1 2oex 8529 . . 3 2o ∈ V
21prid2 4788 . 2 2o ∈ {1o, 2o}
3 onnog 43332 . 2 ((𝐴 ∈ On ∧ 2o ∈ {1o, 2o}) → (𝐴 × {2o}) ∈ No )
42, 3mpan2 690 1 (𝐴 ∈ On → (𝐴 × {2o}) ∈ No )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2103  {csn 4648  {cpr 4650   × cxp 5697  Oncon0 6394  1oc1o 8511  2oc2o 8512   No csur 27693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pow 5386  ax-pr 5450  ax-un 7766
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5170  df-opab 5232  df-mpt 5253  df-id 5597  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-suc 6400  df-fun 6574  df-fn 6575  df-f 6576  df-1o 8518  df-2o 8519  df-no 27696
This theorem is referenced by:  onnoi  43337
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