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Theorem nofv 32399
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 pm2.1 883 . . 3 𝑋 ∈ dom 𝐴𝑋 ∈ dom 𝐴)
2 ndmfv 6476 . . . . 5 𝑋 ∈ dom 𝐴 → (𝐴𝑋) = ∅)
32a1i 11 . . . 4 (𝐴 No → (¬ 𝑋 ∈ dom 𝐴 → (𝐴𝑋) = ∅))
4 nofun 32391 . . . . 5 (𝐴 No → Fun 𝐴)
5 norn 32393 . . . . 5 (𝐴 No → ran 𝐴 ⊆ {1o, 2o})
6 fvelrn 6616 . . . . . . . 8 ((Fun 𝐴𝑋 ∈ dom 𝐴) → (𝐴𝑋) ∈ ran 𝐴)
7 ssel 3815 . . . . . . . 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 445 . . . . . 6 ((Fun 𝐴 ∧ ran 𝐴 ⊆ {1o, 2o}) → (𝑋 ∈ dom 𝐴 → (𝐴𝑋) ∈ {1o, 2o}))
10 1oex 7851 . . . . . . 7 1o ∈ V
11 2on 7852 . . . . . . . 8 2o ∈ On
1211elexi 3415 . . . . . . 7 2o ∈ V
1310, 12elpr2 4422 . . . . . 6 ((𝐴𝑋) ∈ {1o, 2o} ↔ ((𝐴𝑋) = 1o ∨ (𝐴𝑋) = 2o))
149, 13syl6ib 243 . . . . 5 ((Fun 𝐴 ∧ ran 𝐴 ⊆ {1o, 2o}) → (𝑋 ∈ dom 𝐴 → ((𝐴𝑋) = 1o ∨ (𝐴𝑋) = 2o)))
154, 5, 14syl2anc 579 . . . 4 (𝐴 No → (𝑋 ∈ dom 𝐴 → ((𝐴𝑋) = 1o ∨ (𝐴𝑋) = 2o)))
163, 15orim12d 950 . . 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 1074 . 2 (((𝐴𝑋) = ∅ ∨ (𝐴𝑋) = 1o ∨ (𝐴𝑋) = 2o) ↔ ((𝐴𝑋) = ∅ ∨ ((𝐴𝑋) = 1o ∨ (𝐴𝑋) = 2o)))
1917, 18sylibr 226 1 (𝐴 No → ((𝐴𝑋) = ∅ ∨ (𝐴𝑋) = 1o ∨ (𝐴𝑋) = 2o))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 386  wo 836  w3o 1070   = wceq 1601  wcel 2107  wss 3792  c0 4141  {cpr 4400  dom cdm 5355  ran crn 5356  Oncon0 5976  Fun wfun 6129  cfv 6135  1oc1o 7836  2oc2o 7837   No csur 32382
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-ord 5979  df-on 5980  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-1o 7843  df-2o 7844  df-no 32385
This theorem is referenced by:  nolesgn2o  32413  nosep1o  32421  nolt02o  32434  nosupbnd1lem5  32447  nosupbnd1lem6  32448
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