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Theorem mfsdisj 32694
Description: The constants and variables of a formal system are disjoint. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mfsdisj.c 𝐶 = (mCN‘𝑇)
mfsdisj.v 𝑉 = (mVR‘𝑇)
Assertion
Ref Expression
mfsdisj (𝑇 ∈ mFS → (𝐶𝑉) = ∅)

Proof of Theorem mfsdisj
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 mfsdisj.c . . . 4 𝐶 = (mCN‘𝑇)
2 mfsdisj.v . . . 4 𝑉 = (mVR‘𝑇)
3 eqid 2818 . . . 4 (mType‘𝑇) = (mType‘𝑇)
4 eqid 2818 . . . 4 (mVT‘𝑇) = (mVT‘𝑇)
5 eqid 2818 . . . 4 (mTC‘𝑇) = (mTC‘𝑇)
6 eqid 2818 . . . 4 (mAx‘𝑇) = (mAx‘𝑇)
7 eqid 2818 . . . 4 (mStat‘𝑇) = (mStat‘𝑇)
81, 2, 3, 4, 5, 6, 7ismfs 32693 . . 3 (𝑇 ∈ mFS → (𝑇 ∈ mFS ↔ (((𝐶𝑉) = ∅ ∧ (mType‘𝑇):𝑉⟶(mTC‘𝑇)) ∧ ((mAx‘𝑇) ⊆ (mStat‘𝑇) ∧ ∀𝑣 ∈ (mVT‘𝑇) ¬ ((mType‘𝑇) “ {𝑣}) ∈ Fin))))
98ibi 268 . 2 (𝑇 ∈ mFS → (((𝐶𝑉) = ∅ ∧ (mType‘𝑇):𝑉⟶(mTC‘𝑇)) ∧ ((mAx‘𝑇) ⊆ (mStat‘𝑇) ∧ ∀𝑣 ∈ (mVT‘𝑇) ¬ ((mType‘𝑇) “ {𝑣}) ∈ Fin)))
109simplld 764 1 (𝑇 ∈ mFS → (𝐶𝑉) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1528  wcel 2105  wral 3135  cin 3932  wss 3933  c0 4288  {csn 4557  ccnv 5547  cima 5551  wf 6344  cfv 6348  Fincfn 8497  mCNcmcn 32604  mVRcmvar 32605  mTypecmty 32606  mVTcmvt 32607  mTCcmtc 32608  mAxcmax 32609  mStatcmsta 32619  mFScmfs 32620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-mfs 32640
This theorem is referenced by: (None)
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