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Theorem norn 27711
Description: The range of a surreal is a subset of the surreal signs. (Contributed by Scott Fenton, 16-Jun-2011.)
Assertion
Ref Expression
norn (𝐴 No → ran 𝐴 ⊆ {1o, 2o})

Proof of Theorem norn
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elno 27705 . 2 (𝐴 No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o})
2 frn 6744 . . 3 (𝐴:𝑥⟶{1o, 2o} → ran 𝐴 ⊆ {1o, 2o})
32rexlimivw 3149 . 2 (∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o} → ran 𝐴 ⊆ {1o, 2o})
41, 3sylbi 217 1 (𝐴 No → ran 𝐴 ⊆ {1o, 2o})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  wrex 3068  wss 3963  {cpr 4633  ran crn 5690  Oncon0 6386  wf 6559  1oc1o 8498  2oc2o 8499   No csur 27699
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-fun 6565  df-fn 6566  df-f 6567  df-no 27702
This theorem is referenced by:  elno2  27714  nofv  27717  sltres  27722  noextend  27726  noextendseq  27727  nosepssdm  27746  nodenselem8  27751  nolt02olem  27754  nosupno  27763  noinfno  27778
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