Theorem norn 28008

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.

Step	Hyp	Ref	Expression
1		elno 28003	. 2 ⊢ (𝐴 ∈ No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o})
2		frn 6717	. . 3 ⊢ (𝐴:𝑥⟶{1o, 2o} → ran 𝐴 ⊆ {1o, 2o})
3	2	rexlimivw 3159	. 2 ⊢ (∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o} → ran 𝐴 ⊆ {1o, 2o})
4	1, 3	sylbi 220	1 ⊢ (𝐴 ∈ No → ran 𝐴 ⊆ {1o, 2o})

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2144  ∃wrex 3086   ⊆ wss 3895  {cpr 4585  ran crn 5651  Oncon0 6362  ⟶wf 6534  1_oc1o 8469  2_oc2o 8470  Nocsur 27997

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-fun 6540  df-fn 6541  df-f 6542  df-no 28000

This theorem is used by:  elno2  28011  nofv  28014  ltsres  28019  noextend  28023  noextendseq  28024  nosepssdm  28043  nodenselem8  28048  nolt02olem  28051  nosupno  28060  noinfno  28075