Theorem nofv 27591

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 → ((𝐴‘𝑋) = ∅ ∨ (𝐴‘𝑋) = 1o ∨ (𝐴‘𝑋) = 2o))

Proof of Theorem nofv

Step	Hyp	Ref	Expression
1		pm2.1 896	. . 3 ⊢ (¬ 𝑋 ∈ dom 𝐴 ∨ 𝑋 ∈ dom 𝐴)
2		ndmfv 6849	. . . . 5 ⊢ (¬ 𝑋 ∈ dom 𝐴 → (𝐴‘𝑋) = ∅)
3	2	a1i 11	. . . 4 ⊢ (𝐴 ∈ No → (¬ 𝑋 ∈ dom 𝐴 → (𝐴‘𝑋) = ∅))
4		nofun 27583	. . . . 5 ⊢ (𝐴 ∈ No → Fun 𝐴)
5		norn 27585	. . . . 5 ⊢ (𝐴 ∈ No → ran 𝐴 ⊆ {1o, 2o})
6		fvelrn 7004	. . . . . . . 8 ⊢ ((Fun 𝐴 ∧ 𝑋 ∈ dom 𝐴) → (𝐴‘𝑋) ∈ ran 𝐴)
7		ssel 3923	. . . . . . . 8 ⊢ (ran 𝐴 ⊆ {1o, 2o} → ((𝐴‘𝑋) ∈ ran 𝐴 → (𝐴‘𝑋) ∈ {1o, 2o}))
8	6, 7	syl5com 31	. . . . . . 7 ⊢ ((Fun 𝐴 ∧ 𝑋 ∈ dom 𝐴) → (ran 𝐴 ⊆ {1o, 2o} → (𝐴‘𝑋) ∈ {1o, 2o}))
9	8	impancom 451	. . . . . 6 ⊢ ((Fun 𝐴 ∧ ran 𝐴 ⊆ {1o, 2o}) → (𝑋 ∈ dom 𝐴 → (𝐴‘𝑋) ∈ {1o, 2o}))
10		1oex 8390	. . . . . . 7 ⊢ 1o ∈ V
11		2on 8393	. . . . . . . 8 ⊢ 2o ∈ On
12	11	elexi 3459	. . . . . . 7 ⊢ 2o ∈ V
13	10, 12	elpr2 4598	. . . . . 6 ⊢ ((𝐴‘𝑋) ∈ {1o, 2o} ↔ ((𝐴‘𝑋) = 1o ∨ (𝐴‘𝑋) = 2o))
14	9, 13	imbitrdi 251	. . . . 5 ⊢ ((Fun 𝐴 ∧ ran 𝐴 ⊆ {1o, 2o}) → (𝑋 ∈ dom 𝐴 → ((𝐴‘𝑋) = 1o ∨ (𝐴‘𝑋) = 2o)))
15	4, 5, 14	syl2anc 584	. . . 4 ⊢ (𝐴 ∈ No → (𝑋 ∈ dom 𝐴 → ((𝐴‘𝑋) = 1o ∨ (𝐴‘𝑋) = 2o)))
16	3, 15	orim12d 966	. . 3 ⊢ (𝐴 ∈ No → ((¬ 𝑋 ∈ dom 𝐴 ∨ 𝑋 ∈ dom 𝐴) → ((𝐴‘𝑋) = ∅ ∨ ((𝐴‘𝑋) = 1o ∨ (𝐴‘𝑋) = 2o))))
17	1, 16	mpi 20	. 2 ⊢ (𝐴 ∈ No → ((𝐴‘𝑋) = ∅ ∨ ((𝐴‘𝑋) = 1o ∨ (𝐴‘𝑋) = 2o)))
18		3orass 1089	. 2 ⊢ (((𝐴‘𝑋) = ∅ ∨ (𝐴‘𝑋) = 1o ∨ (𝐴‘𝑋) = 2o) ↔ ((𝐴‘𝑋) = ∅ ∨ ((𝐴‘𝑋) = 1o ∨ (𝐴‘𝑋) = 2o)))
19	17, 18	sylibr 234	1 ⊢ (𝐴 ∈ No → ((𝐴‘𝑋) = ∅ ∨ (𝐴‘𝑋) = 1o ∨ (𝐴‘𝑋) = 2o))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 847   ∨ w3o 1085   = wceq 1541   ∈ wcel 2111   ⊆ wss 3897  ∅c0 4278  {cpr 4573  dom cdm 5611  ran crn 5612  Oncon0 6301  Fun wfun 6470  ‘cfv 6476  1_oc1o 8373  2_oc2o 8374   No csur 27573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-tr 5194  df-id 5506  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-we 5566  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-ord 6304  df-on 6305  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-fv 6484  df-1o 8380  df-2o 8381  df-no 27576

This theorem is referenced by:  nolesgn2o  27605  nogesgn1o  27607  nosep1o  27615  nosep2o  27616  nolt02o  27629  nogt01o  27630  nosupbnd1lem5  27646  nosupbnd1lem6  27647  noinfbnd1lem5  27661  noinfbnd1lem6  27662