MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  norn Structured version   Visualization version   GIF version

Theorem norn 27696
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 27690 . 2 (𝐴 No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o})
2 frn 6743 . . 3 (𝐴:𝑥⟶{1o, 2o} → ran 𝐴 ⊆ {1o, 2o})
32rexlimivw 3151 . 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 2108  wrex 3070  wss 3951  {cpr 4628  ran crn 5686  Oncon0 6384  wf 6557  1oc1o 8499  2oc2o 8500   No csur 27684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-fun 6563  df-fn 6564  df-f 6565  df-no 27687
This theorem is referenced by:  elno2  27699  nofv  27702  sltres  27707  noextend  27711  noextendseq  27712  nosepssdm  27731  nodenselem8  27736  nolt02olem  27739  nosupno  27748  noinfno  27763
  Copyright terms: Public domain W3C validator