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Theorem norn 31048
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 31043 . 2 (𝐴 No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1𝑜, 2𝑜})
2 frn 5966 . . 3 (𝐴:𝑥⟶{1𝑜, 2𝑜} → ran 𝐴 ⊆ {1𝑜, 2𝑜})
32rexlimivw 3011 . 2 (∃𝑥 ∈ On 𝐴:𝑥⟶{1𝑜, 2𝑜} → ran 𝐴 ⊆ {1𝑜, 2𝑜})
41, 3sylbi 206 1 (𝐴 No → ran 𝐴 ⊆ {1𝑜, 2𝑜})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1977  wrex 2897  wss 3540  {cpr 4127  ran crn 5039  Oncon0 5640  wf 5800  1𝑜c1o 7440  2𝑜c2o 7441   No csur 31037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-no 31040
This theorem is referenced by:  elno2  31051  nofv  31054  sltres  31061  nodenselem3  31082  nodenselem6  31085  nodenselem8  31087
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