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Theorem onnog 41605
Description: Every ordinal maps to a surreal number. (Contributed by RP, 21-Sep-2023.)
Assertion
Ref Expression
onnog ((𝐴 ∈ On ∧ 𝐵 ∈ {1o, 2o}) → (𝐴 × {𝐵}) ∈ No )

Proof of Theorem onnog
StepHypRef Expression
1 fconst6g 6728 . . 3 (𝐵 ∈ {1o, 2o} → (𝐴 × {𝐵}):𝐴⟶{1o, 2o})
21adantl 482 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ {1o, 2o}) → (𝐴 × {𝐵}):𝐴⟶{1o, 2o})
3 simp3 1138 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ {1o, 2o} ∧ (𝐴 × {𝐵}):𝐴⟶{1o, 2o}) → (𝐴 × {𝐵}):𝐴⟶{1o, 2o})
43ffund 6669 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ {1o, 2o} ∧ (𝐴 × {𝐵}):𝐴⟶{1o, 2o}) → Fun (𝐴 × {𝐵}))
5 simp2 1137 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ {1o, 2o} ∧ (𝐴 × {𝐵}):𝐴⟶{1o, 2o}) → 𝐵 ∈ {1o, 2o})
6 snnzg 4733 . . . . 5 (𝐵 ∈ {1o, 2o} → {𝐵} ≠ ∅)
7 dmxp 5882 . . . . . 6 ({𝐵} ≠ ∅ → dom (𝐴 × {𝐵}) = 𝐴)
87eqcomd 2743 . . . . 5 ({𝐵} ≠ ∅ → 𝐴 = dom (𝐴 × {𝐵}))
95, 6, 83syl 18 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ {1o, 2o} ∧ (𝐴 × {𝐵}):𝐴⟶{1o, 2o}) → 𝐴 = dom (𝐴 × {𝐵}))
10 simp1 1136 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ {1o, 2o} ∧ (𝐴 × {𝐵}):𝐴⟶{1o, 2o}) → 𝐴 ∈ On)
119, 10eqeltrrd 2839 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ {1o, 2o} ∧ (𝐴 × {𝐵}):𝐴⟶{1o, 2o}) → dom (𝐴 × {𝐵}) ∈ On)
123frnd 6673 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ {1o, 2o} ∧ (𝐴 × {𝐵}):𝐴⟶{1o, 2o}) → ran (𝐴 × {𝐵}) ⊆ {1o, 2o})
13 elno2 26953 . . 3 ((𝐴 × {𝐵}) ∈ No ↔ (Fun (𝐴 × {𝐵}) ∧ dom (𝐴 × {𝐵}) ∈ On ∧ ran (𝐴 × {𝐵}) ⊆ {1o, 2o}))
144, 11, 12, 13syl3anbrc 1343 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ {1o, 2o} ∧ (𝐴 × {𝐵}):𝐴⟶{1o, 2o}) → (𝐴 × {𝐵}) ∈ No )
152, 14mpd3an3 1462 1 ((𝐴 ∈ On ∧ 𝐵 ∈ {1o, 2o}) → (𝐴 × {𝐵}) ∈ No )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2941  wss 3908  c0 4280  {csn 4584  {cpr 4586   × cxp 5629  dom cdm 5631  ran crn 5632  Oncon0 6315  Fun wfun 6487  wf 6489  1oc1o 8397  2oc2o 8398   No csur 26939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-no 26942
This theorem is referenced by:  onnobdayg  41606  bdaybndex  41607  onno  41609
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