Theorem mfsdisj 35534

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 2734	. . . 4 ⊢ (mType‘𝑇) = (mType‘𝑇)
4		eqid 2734	. . . 4 ⊢ (mVT‘𝑇) = (mVT‘𝑇)
5		eqid 2734	. . . 4 ⊢ (mTC‘𝑇) = (mTC‘𝑇)
6		eqid 2734	. . . 4 ⊢ (mAx‘𝑇) = (mAx‘𝑇)
7		eqid 2734	. . . 4 ⊢ (mStat‘𝑇) = (mStat‘𝑇)
8	1, 2, 3, 4, 5, 6, 7	ismfs 35533	. . 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 768	1 ⊢ (𝑇 ∈ mFS → (𝐶 ∩ 𝑉) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1536   ∈ wcel 2105  ∀wral 3058   ∩ cin 3961   ⊆ wss 3962  ∅c0 4338  {csn 4630  ^◡ccnv 5687   “ cima 5691  ⟶wf 6558  ‘cfv 6562  Fincfn 8983  mCNcmcn 35444  mVRcmvar 35445  mTypecmty 35446  mVTcmvt 35447  mTCcmtc 35448  mAxcmax 35449  mStatcmsta 35459  mFScmfs 35460

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fv 6570  df-mfs 35480

This theorem is referenced by: (None)