Theorem mfsdisj 35527

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 2729	. . . 4 ⊢ (mType‘𝑇) = (mType‘𝑇)
4		eqid 2729	. . . 4 ⊢ (mVT‘𝑇) = (mVT‘𝑇)
5		eqid 2729	. . . 4 ⊢ (mTC‘𝑇) = (mTC‘𝑇)
6		eqid 2729	. . . 4 ⊢ (mAx‘𝑇) = (mAx‘𝑇)
7		eqid 2729	. . . 4 ⊢ (mStat‘𝑇) = (mStat‘𝑇)
8	1, 2, 3, 4, 5, 6, 7	ismfs 35526	. . 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 267	. 2 ⊢ (𝑇 ∈ mFS → (((𝐶 ∩ 𝑉) = ∅ ∧ (mType‘𝑇):𝑉⟶(mTC‘𝑇)) ∧ ((mAx‘𝑇) ⊆ (mStat‘𝑇) ∧ ∀𝑣 ∈ (mVT‘𝑇) ¬ (◡(mType‘𝑇) “ {𝑣}) ∈ Fin)))
10	9	simplld 767	1 ⊢ (𝑇 ∈ mFS → (𝐶 ∩ 𝑉) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044   ∩ cin 3902   ⊆ wss 3903  ∅c0 4284  {csn 4577  ^◡ccnv 5618   “ cima 5622  ⟶wf 6478  ‘cfv 6482  Fincfn 8872  mCNcmcn 35437  mVRcmvar 35438  mTypecmty 35439  mVTcmvt 35440  mTCcmtc 35441  mAxcmax 35442  mStatcmsta 35452  mFScmfs 35453

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-fv 6490  df-mfs 35473

This theorem is referenced by: (None)