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Theorem nofv 32141
 Description: The function value of a surreal is either a sign or the empty set. (Contributed by Scott Fenton, 22-Jun-2011.)
Assertion
Ref Expression
nofv (𝐴 No → ((𝐴𝑋) = ∅ ∨ (𝐴𝑋) = 1𝑜 ∨ (𝐴𝑋) = 2𝑜))

Proof of Theorem nofv
StepHypRef Expression
1 pm2.1 862 . . 3 𝑋 ∈ dom 𝐴𝑋 ∈ dom 𝐴)
2 ndmfv 6359 . . . . 5 𝑋 ∈ dom 𝐴 → (𝐴𝑋) = ∅)
32a1i 11 . . . 4 (𝐴 No → (¬ 𝑋 ∈ dom 𝐴 → (𝐴𝑋) = ∅))
4 nofun 32133 . . . . 5 (𝐴 No → Fun 𝐴)
5 norn 32135 . . . . 5 (𝐴 No → ran 𝐴 ⊆ {1𝑜, 2𝑜})
6 fvelrn 6495 . . . . . . . 8 ((Fun 𝐴𝑋 ∈ dom 𝐴) → (𝐴𝑋) ∈ ran 𝐴)
7 ssel 3744 . . . . . . . 8 (ran 𝐴 ⊆ {1𝑜, 2𝑜} → ((𝐴𝑋) ∈ ran 𝐴 → (𝐴𝑋) ∈ {1𝑜, 2𝑜}))
86, 7syl5com 31 . . . . . . 7 ((Fun 𝐴𝑋 ∈ dom 𝐴) → (ran 𝐴 ⊆ {1𝑜, 2𝑜} → (𝐴𝑋) ∈ {1𝑜, 2𝑜}))
98impancom 439 . . . . . 6 ((Fun 𝐴 ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜}) → (𝑋 ∈ dom 𝐴 → (𝐴𝑋) ∈ {1𝑜, 2𝑜}))
10 1oex 7720 . . . . . . 7 1𝑜 ∈ V
11 2on 7721 . . . . . . . 8 2𝑜 ∈ On
1211elexi 3362 . . . . . . 7 2𝑜 ∈ V
1310, 12elpr2 4337 . . . . . 6 ((𝐴𝑋) ∈ {1𝑜, 2𝑜} ↔ ((𝐴𝑋) = 1𝑜 ∨ (𝐴𝑋) = 2𝑜))
149, 13syl6ib 241 . . . . 5 ((Fun 𝐴 ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜}) → (𝑋 ∈ dom 𝐴 → ((𝐴𝑋) = 1𝑜 ∨ (𝐴𝑋) = 2𝑜)))
154, 5, 14syl2anc 565 . . . 4 (𝐴 No → (𝑋 ∈ dom 𝐴 → ((𝐴𝑋) = 1𝑜 ∨ (𝐴𝑋) = 2𝑜)))
163, 15orim12d 945 . . 3 (𝐴 No → ((¬ 𝑋 ∈ dom 𝐴𝑋 ∈ dom 𝐴) → ((𝐴𝑋) = ∅ ∨ ((𝐴𝑋) = 1𝑜 ∨ (𝐴𝑋) = 2𝑜))))
171, 16mpi 20 . 2 (𝐴 No → ((𝐴𝑋) = ∅ ∨ ((𝐴𝑋) = 1𝑜 ∨ (𝐴𝑋) = 2𝑜)))
18 3orass 1073 . 2 (((𝐴𝑋) = ∅ ∨ (𝐴𝑋) = 1𝑜 ∨ (𝐴𝑋) = 2𝑜) ↔ ((𝐴𝑋) = ∅ ∨ ((𝐴𝑋) = 1𝑜 ∨ (𝐴𝑋) = 2𝑜)))
1917, 18sylibr 224 1 (𝐴 No → ((𝐴𝑋) = ∅ ∨ (𝐴𝑋) = 1𝑜 ∨ (𝐴𝑋) = 2𝑜))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 382   ∨ wo 826   ∨ w3o 1069   = wceq 1630   ∈ wcel 2144   ⊆ wss 3721  ∅c0 4061  {cpr 4316  dom cdm 5249  ran crn 5250  Oncon0 5866  Fun wfun 6025  ‘cfv 6031  1𝑜c1o 7705  2𝑜c2o 7706   No csur 32124 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-ord 5869  df-on 5870  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-1o 7712  df-2o 7713  df-no 32127 This theorem is referenced by:  nolesgn2o  32155  nosep1o  32163  nolt02o  32176  nosupbnd1lem5  32189  nosupbnd1lem6  32190
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