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Theorem nofv 27703
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 pm2.1 896 . . 3 𝑋 ∈ dom 𝐴𝑋 ∈ dom 𝐴)
2 ndmfv 6940 . . . . 5 𝑋 ∈ dom 𝐴 → (𝐴𝑋) = ∅)
32a1i 11 . . . 4 (𝐴 No → (¬ 𝑋 ∈ dom 𝐴 → (𝐴𝑋) = ∅))
4 nofun 27695 . . . . 5 (𝐴 No → Fun 𝐴)
5 norn 27697 . . . . 5 (𝐴 No → ran 𝐴 ⊆ {1o, 2o})
6 fvelrn 7095 . . . . . . . 8 ((Fun 𝐴𝑋 ∈ dom 𝐴) → (𝐴𝑋) ∈ ran 𝐴)
7 ssel 3976 . . . . . . . 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 451 . . . . . 6 ((Fun 𝐴 ∧ ran 𝐴 ⊆ {1o, 2o}) → (𝑋 ∈ dom 𝐴 → (𝐴𝑋) ∈ {1o, 2o}))
10 1oex 8517 . . . . . . 7 1o ∈ V
11 2on 8521 . . . . . . . 8 2o ∈ On
1211elexi 3502 . . . . . . 7 2o ∈ V
1310, 12elpr2 4651 . . . . . 6 ((𝐴𝑋) ∈ {1o, 2o} ↔ ((𝐴𝑋) = 1o ∨ (𝐴𝑋) = 2o))
149, 13imbitrdi 251 . . . . 5 ((Fun 𝐴 ∧ ran 𝐴 ⊆ {1o, 2o}) → (𝑋 ∈ dom 𝐴 → ((𝐴𝑋) = 1o ∨ (𝐴𝑋) = 2o)))
154, 5, 14syl2anc 584 . . . 4 (𝐴 No → (𝑋 ∈ dom 𝐴 → ((𝐴𝑋) = 1o ∨ (𝐴𝑋) = 2o)))
163, 15orim12d 966 . . 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 1089 . 2 (((𝐴𝑋) = ∅ ∨ (𝐴𝑋) = 1o ∨ (𝐴𝑋) = 2o) ↔ ((𝐴𝑋) = ∅ ∨ ((𝐴𝑋) = 1o ∨ (𝐴𝑋) = 2o)))
1917, 18sylibr 234 1 (𝐴 No → ((𝐴𝑋) = ∅ ∨ (𝐴𝑋) = 1o ∨ (𝐴𝑋) = 2o))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 847  w3o 1085   = wceq 1539  wcel 2107  wss 3950  c0 4332  {cpr 4627  dom cdm 5684  ran crn 5685  Oncon0 6383  Fun wfun 6554  cfv 6560  1oc1o 8500  2oc2o 8501   No csur 27685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-ord 6386  df-on 6387  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fv 6568  df-1o 8507  df-2o 8508  df-no 27688
This theorem is referenced by:  nolesgn2o  27717  nogesgn1o  27719  nosep1o  27727  nosep2o  27728  nolt02o  27741  nogt01o  27742  nosupbnd1lem5  27758  nosupbnd1lem6  27759  noinfbnd1lem5  27773  noinfbnd1lem6  27774
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