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Theorem nofv 31564
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 pm2.1 433 . . 3 𝑋 ∈ dom 𝐴𝑋 ∈ dom 𝐴)
2 ndmfv 6185 . . . . 5 𝑋 ∈ dom 𝐴 → (𝐴𝑋) = ∅)
32a1i 11 . . . 4 (𝐴 No → (¬ 𝑋 ∈ dom 𝐴 → (𝐴𝑋) = ∅))
4 nofun 31556 . . . . 5 (𝐴 No → Fun 𝐴)
5 norn 31558 . . . . 5 (𝐴 No → ran 𝐴 ⊆ {1𝑜, 2𝑜})
6 fvelrn 6318 . . . . . . . 8 ((Fun 𝐴𝑋 ∈ dom 𝐴) → (𝐴𝑋) ∈ ran 𝐴)
7 ssel 3582 . . . . . . . 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 456 . . . . . 6 ((Fun 𝐴 ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜}) → (𝑋 ∈ dom 𝐴 → (𝐴𝑋) ∈ {1𝑜, 2𝑜}))
10 1on 7527 . . . . . . . 8 1𝑜 ∈ On
1110elexi 3203 . . . . . . 7 1𝑜 ∈ V
12 2on 7528 . . . . . . . 8 2𝑜 ∈ On
1312elexi 3203 . . . . . . 7 2𝑜 ∈ V
1411, 13elpr2 4177 . . . . . 6 ((𝐴𝑋) ∈ {1𝑜, 2𝑜} ↔ ((𝐴𝑋) = 1𝑜 ∨ (𝐴𝑋) = 2𝑜))
159, 14syl6ib 241 . . . . 5 ((Fun 𝐴 ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜}) → (𝑋 ∈ dom 𝐴 → ((𝐴𝑋) = 1𝑜 ∨ (𝐴𝑋) = 2𝑜)))
164, 5, 15syl2anc 692 . . . 4 (𝐴 No → (𝑋 ∈ dom 𝐴 → ((𝐴𝑋) = 1𝑜 ∨ (𝐴𝑋) = 2𝑜)))
173, 16orim12d 882 . . 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 1039 . 2 (((𝐴𝑋) = ∅ ∨ (𝐴𝑋) = 1𝑜 ∨ (𝐴𝑋) = 2𝑜) ↔ ((𝐴𝑋) = ∅ ∨ ((𝐴𝑋) = 1𝑜 ∨ (𝐴𝑋) = 2𝑜)))
2018, 19sylibr 224 1 (𝐴 No → ((𝐴𝑋) = ∅ ∨ (𝐴𝑋) = 1𝑜 ∨ (𝐴𝑋) = 2𝑜))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 383  wa 384  w3o 1035   = wceq 1480  wcel 1987  wss 3560  c0 3897  {cpr 4157  dom cdm 5084  ran crn 5085  Oncon0 5692  Fun wfun 5851  cfv 5857  1𝑜c1o 7513  2𝑜c2o 7514   No csur 31547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-ord 5695  df-on 5696  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-1o 7520  df-2o 7521  df-no 31550
This theorem is referenced by:  nobndup  31616  nobnddown  31617
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