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Theorem nofv 31508
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 6175 . . . . 5 𝑋 ∈ dom 𝐴 → (𝐴𝑋) = ∅)
32a1i 11 . . . 4 (𝐴 No → (¬ 𝑋 ∈ dom 𝐴 → (𝐴𝑋) = ∅))
4 nofun 31500 . . . . 5 (𝐴 No → Fun 𝐴)
5 norn 31502 . . . . 5 (𝐴 No → ran 𝐴 ⊆ {1𝑜, 2𝑜})
6 fvelrn 6308 . . . . . . . 8 ((Fun 𝐴𝑋 ∈ dom 𝐴) → (𝐴𝑋) ∈ ran 𝐴)
7 ssel 3577 . . . . . . . 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 7512 . . . . . . . 8 1𝑜 ∈ On
1110elexi 3199 . . . . . . 7 1𝑜 ∈ V
12 2on 7513 . . . . . . . 8 2𝑜 ∈ On
1312elexi 3199 . . . . . . 7 2𝑜 ∈ V
1411, 13elpr2 4170 . . . . . 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 3555  c0 3891  {cpr 4150  dom cdm 5074  ran crn 5075  Oncon0 5682  Fun wfun 5841  cfv 5847  1𝑜c1o 7498  2𝑜c2o 7499   No csur 31491
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 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
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 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-ord 5685  df-on 5686  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-1o 7505  df-2o 7506  df-no 31494
This theorem is referenced by:  nobndup  31560  nobnddown  31561
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