Theorem mfsdisj 36063

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 → (𝐶 ∩ 𝑉) = ∅)

Proof of Theorem mfsdisj

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mfsdisj.c	. . . 4 ⊢ 𝐶 = (mCN‘𝑇)
2		mfsdisj.v	. . . 4 ⊢ 𝑉 = (mVR‘𝑇)
3		eqid 2766	. . . 4 ⊢ (mType‘𝑇) = (mType‘𝑇)
4		eqid 2766	. . . 4 ⊢ (mVT‘𝑇) = (mVT‘𝑇)
5		eqid 2766	. . . 4 ⊢ (mTC‘𝑇) = (mTC‘𝑇)
6		eqid 2766	. . . 4 ⊢ (mAx‘𝑇) = (mAx‘𝑇)
7		eqid 2766	. . . 4 ⊢ (mStat‘𝑇) = (mStat‘𝑇)
8	1, 2, 3, 4, 5, 6, 7	ismfs 36062	. . 3 ⊢ (𝑇 ∈ mFS → (𝑇 ∈ mFS ↔ (((𝐶 ∩ 𝑉) = ∅ ∧ (mType‘𝑇):𝑉⟶(mTC‘𝑇)) ∧ ((mAx‘𝑇) ⊆ (mStat‘𝑇) ∧ ∀𝑣 ∈ (mVT‘𝑇) ¬ (◡(mType‘𝑇) “ {𝑣}) ∈ Fin))))
9	8	ibi 270	. 2 ⊢ (𝑇 ∈ mFS → (((𝐶 ∩ 𝑉) = ∅ ∧ (mType‘𝑇):𝑉⟶(mTC‘𝑇)) ∧ ((mAx‘𝑇) ⊆ (mStat‘𝑇) ∧ ∀𝑣 ∈ (mVT‘𝑇) ¬ (◡(mType‘𝑇) “ {𝑣}) ∈ Fin)))
10	9	simplld 780	1 ⊢ (𝑇 ∈ mFS → (𝐶 ∩ 𝑉) = ∅)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3082   ∩ cin 3907   ⊆ wss 3908  ∅c0 4289  {csn 4594  ^◡ccnv 5665   “ cima 5669  ⟶wf 6539  ‘cfv 6543  Fincfn 8952  mCNcmcn 35973  mVRcmvar 35974  mTypecmty 35975  mVTcmvt 35976  mTCcmtc 35977  mAxcmax 35978  mStatcmsta 35988  mFScmfs 35989

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-mfs 36009

This theorem is used by: (None)