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Theorem mfsdisj 32785
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 2819 . . . 4 (mType‘𝑇) = (mType‘𝑇)
4 eqid 2819 . . . 4 (mVT‘𝑇) = (mVT‘𝑇)
5 eqid 2819 . . . 4 (mTC‘𝑇) = (mTC‘𝑇)
6 eqid 2819 . . . 4 (mAx‘𝑇) = (mAx‘𝑇)
7 eqid 2819 . . . 4 (mStat‘𝑇) = (mStat‘𝑇)
81, 2, 3, 4, 5, 6, 7ismfs 32784 . . 3 (𝑇 ∈ mFS → (𝑇 ∈ mFS ↔ (((𝐶𝑉) = ∅ ∧ (mType‘𝑇):𝑉⟶(mTC‘𝑇)) ∧ ((mAx‘𝑇) ⊆ (mStat‘𝑇) ∧ ∀𝑣 ∈ (mVT‘𝑇) ¬ ((mType‘𝑇) “ {𝑣}) ∈ Fin))))
98ibi 269 . 2 (𝑇 ∈ mFS → (((𝐶𝑉) = ∅ ∧ (mType‘𝑇):𝑉⟶(mTC‘𝑇)) ∧ ((mAx‘𝑇) ⊆ (mStat‘𝑇) ∧ ∀𝑣 ∈ (mVT‘𝑇) ¬ ((mType‘𝑇) “ {𝑣}) ∈ Fin)))
109simplld 766 1 (𝑇 ∈ mFS → (𝐶𝑉) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 398   = wceq 1530  wcel 2107  wral 3136  cin 3933  wss 3934  c0 4289  {csn 4559  ccnv 5547  cima 5551  wf 6344  cfv 6348  Fincfn 8501  mCNcmcn 32695  mVRcmvar 32696  mTypecmty 32697  mVTcmvt 32698  mTCcmtc 32699  mAxcmax 32700  mStatcmsta 32710  mFScmfs 32711
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  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 32731
This theorem is referenced by: (None)
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