Theorem norn 28035

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 28030	. 2 ⊢ (𝐴 ∈ No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o})
2		frn 6736	. . 3 ⊢ (𝐴:𝑥⟶{1o, 2o} → ran 𝐴 ⊆ {1o, 2o})
3	2	rexlimivw 3163	. 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 2147  ∃wrex 3090   ⊆ wss 3900  {cpr 4590  ran crn 5656  Oncon0 6377  ⟶wf 6551  1_oc1o 8494  2_oc2o 8495  Nocsur 28024

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pow 5331  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-fun 6557  df-fn 6558  df-f 6559  df-no 28027

This theorem is used by:  elno2  28038  nofv  28041  ltsres  28046  noextend  28050  noextendseq  28051  nosepssdm  28070  nodenselem8  28075  nolt02olem  28078  nosupno  28087  noinfno  28102