Theorem mfsdisj 36401

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 2760	. . . 4 ⊢ (mType‘𝑇) = (mType‘𝑇)
4		eqid 2760	. . . 4 ⊢ (mVT‘𝑇) = (mVT‘𝑇)
5		eqid 2760	. . . 4 ⊢ (mTC‘𝑇) = (mTC‘𝑇)
6		eqid 2760	. . . 4 ⊢ (mAx‘𝑇) = (mAx‘𝑇)
7		eqid 2760	. . . 4 ⊢ (mStat‘𝑇) = (mStat‘𝑇)
8	1, 2, 3, 4, 5, 6, 7	ismfs 36400	. . 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 1569   ∈ wcel 2144  ∀wral 3076   ⊆ wss 3895   ∩ cin 4089  ∅c0 4278  {csn 4584  ^◡ccnv 5649   “ cima 5653  ⟶wf 6534  ‘cfv 6538  Fincfn 8976  mCNcmcn 36311  mVRcmvar 36312  mTypecmty 36313  mVTcmvt 36314  mTCcmtc 36315  mAxcmax 36316  mStatcmsta 36326  mFScmfs 36327

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-mfs 36347

This theorem is used by: (None)