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Theorem nofv 31054
 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 pm2.1 432 . . 3 𝑋 ∈ dom 𝐴𝑋 ∈ dom 𝐴)
2 ndmfv 6128 . . . . 5 𝑋 ∈ dom 𝐴 → (𝐴𝑋) = ∅)
32a1i 11 . . . 4 (𝐴 No → (¬ 𝑋 ∈ dom 𝐴 → (𝐴𝑋) = ∅))
4 nofun 31046 . . . . 5 (𝐴 No → Fun 𝐴)
5 norn 31048 . . . . 5 (𝐴 No → ran 𝐴 ⊆ {1𝑜, 2𝑜})
6 fvelrn 6260 . . . . . . . 8 ((Fun 𝐴𝑋 ∈ dom 𝐴) → (𝐴𝑋) ∈ ran 𝐴)
7 ssel 3562 . . . . . . . 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 455 . . . . . 6 ((Fun 𝐴 ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜}) → (𝑋 ∈ dom 𝐴 → (𝐴𝑋) ∈ {1𝑜, 2𝑜}))
10 1on 7454 . . . . . . . 8 1𝑜 ∈ On
1110elexi 3186 . . . . . . 7 1𝑜 ∈ V
12 2on 7455 . . . . . . . 8 2𝑜 ∈ On
1312elexi 3186 . . . . . . 7 2𝑜 ∈ V
1411, 13elpr2 4147 . . . . . 6 ((𝐴𝑋) ∈ {1𝑜, 2𝑜} ↔ ((𝐴𝑋) = 1𝑜 ∨ (𝐴𝑋) = 2𝑜))
159, 14syl6ib 240 . . . . 5 ((Fun 𝐴 ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜}) → (𝑋 ∈ dom 𝐴 → ((𝐴𝑋) = 1𝑜 ∨ (𝐴𝑋) = 2𝑜)))
164, 5, 15syl2anc 691 . . . 4 (𝐴 No → (𝑋 ∈ dom 𝐴 → ((𝐴𝑋) = 1𝑜 ∨ (𝐴𝑋) = 2𝑜)))
173, 16orim12d 879 . . 3 (𝐴 No → ((¬ 𝑋 ∈ dom 𝐴𝑋 ∈ dom 𝐴) → ((𝐴𝑋) = ∅ ∨ ((𝐴𝑋) = 1𝑜 ∨ (𝐴𝑋) = 2𝑜))))
181, 17mpi 20 . 2 (𝐴 No → ((𝐴𝑋) = ∅ ∨ ((𝐴𝑋) = 1𝑜 ∨ (𝐴𝑋) = 2𝑜)))
19 3orass 1034 . 2 (((𝐴𝑋) = ∅ ∨ (𝐴𝑋) = 1𝑜 ∨ (𝐴𝑋) = 2𝑜) ↔ ((𝐴𝑋) = ∅ ∨ ((𝐴𝑋) = 1𝑜 ∨ (𝐴𝑋) = 2𝑜)))
2018, 19sylibr 223 1 (𝐴 No → ((𝐴𝑋) = ∅ ∨ (𝐴𝑋) = 1𝑜 ∨ (𝐴𝑋) = 2𝑜))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383   ∨ w3o 1030   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540  ∅c0 3874  {cpr 4127  dom cdm 5038  ran crn 5039  Oncon0 5640  Fun wfun 5798  ‘cfv 5804  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-8 1979  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-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  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-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  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-ord 5643  df-on 5644  df-suc 5646  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-1o 7447  df-2o 7448  df-no 31040 This theorem is referenced by:  nobndup  31099  nobnddown  31100
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