Theorem mfsdisj 33618

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 2737	. . . 4 ⊢ (mType‘𝑇) = (mType‘𝑇)
4		eqid 2737	. . . 4 ⊢ (mVT‘𝑇) = (mVT‘𝑇)
5		eqid 2737	. . . 4 ⊢ (mTC‘𝑇) = (mTC‘𝑇)
6		eqid 2737	. . . 4 ⊢ (mAx‘𝑇) = (mAx‘𝑇)
7		eqid 2737	. . . 4 ⊢ (mStat‘𝑇) = (mStat‘𝑇)
8	1, 2, 3, 4, 5, 6, 7	ismfs 33617	. . 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 266	. 2 ⊢ (𝑇 ∈ mFS → (((𝐶 ∩ 𝑉) = ∅ ∧ (mType‘𝑇):𝑉⟶(mTC‘𝑇)) ∧ ((mAx‘𝑇) ⊆ (mStat‘𝑇) ∧ ∀𝑣 ∈ (mVT‘𝑇) ¬ (◡(mType‘𝑇) “ {𝑣}) ∈ Fin)))
10	9	simplld 765	1 ⊢ (𝑇 ∈ mFS → (𝐶 ∩ 𝑉) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1540   ∈ wcel 2105  ∀wral 3062   ∩ cin 3895   ⊆ wss 3896  ∅c0 4266  {csn 4569  ^◡ccnv 5604   “ cima 5608  ⟶wf 6459  ‘cfv 6463  Fincfn 8779  mCNcmcn 33528  mVRcmvar 33529  mTypecmty 33530  mVTcmvt 33531  mTCcmtc 33532  mAxcmax 33533  mStatcmsta 33543  mFScmfs 33544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2708

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3063  df-rab 3405  df-v 3443  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-nul 4267  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4849  df-br 5086  df-opab 5148  df-rel 5612  df-cnv 5613  df-co 5614  df-dm 5615  df-rn 5616  df-res 5617  df-ima 5618  df-iota 6415  df-fun 6465  df-fn 6466  df-f 6467  df-fv 6471  df-mfs 33564

This theorem is referenced by: (None)