Theorem maxsta 36288

Description: An axiom is a statement. (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypotheses

Ref	Expression
maxsta.a	⊢ 𝐴 = (mAx‘𝑇)
maxsta.s	⊢ 𝑆 = (mStat‘𝑇)

Assertion

Ref	Expression
maxsta	⊢ (𝑇 ∈ mFS → 𝐴 ⊆ 𝑆)

Proof of Theorem maxsta

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2761	. . . 4 ⊢ (mCN‘𝑇) = (mCN‘𝑇)
2		eqid 2761	. . . 4 ⊢ (mVR‘𝑇) = (mVR‘𝑇)
3		eqid 2761	. . . 4 ⊢ (mType‘𝑇) = (mType‘𝑇)
4		eqid 2761	. . . 4 ⊢ (mVT‘𝑇) = (mVT‘𝑇)
5		eqid 2761	. . . 4 ⊢ (mTC‘𝑇) = (mTC‘𝑇)
6		maxsta.a	. . . 4 ⊢ 𝐴 = (mAx‘𝑇)
7		maxsta.s	. . . 4 ⊢ 𝑆 = (mStat‘𝑇)
8	1, 2, 3, 4, 5, 6, 7	ismfs 36283	. . 3 ⊢ (𝑇 ∈ mFS → (𝑇 ∈ mFS ↔ ((((mCN‘𝑇) ∩ (mVR‘𝑇)) = ∅ ∧ (mType‘𝑇):(mVR‘𝑇)⟶(mTC‘𝑇)) ∧ (𝐴 ⊆ 𝑆 ∧ ∀𝑣 ∈ (mVT‘𝑇) ¬ (◡(mType‘𝑇) “ {𝑣}) ∈ Fin))))
9	8	ibi 270	. 2 ⊢ (𝑇 ∈ mFS → ((((mCN‘𝑇) ∩ (mVR‘𝑇)) = ∅ ∧ (mType‘𝑇):(mVR‘𝑇)⟶(mTC‘𝑇)) ∧ (𝐴 ⊆ 𝑆 ∧ ∀𝑣 ∈ (mVT‘𝑇) ¬ (◡(mType‘𝑇) “ {𝑣}) ∈ Fin)))
10	9	simprld 784	1 ⊢ (𝑇 ∈ mFS → 𝐴 ⊆ 𝑆)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279  {csn 4584  ^◡ccnv 5650   “ cima 5654  ⟶wf 6527  ‘cfv 6531  Fincfn 8957  mCNcmcn 36194  mVRcmvar 36195  mTypecmty 36196  mVTcmvt 36197  mTCcmtc 36198  mAxcmax 36199  mStatcmsta 36209  mFScmfs 36210

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 2147  ax-9 2155  ax-ext 2733

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 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-fv 6539  df-mfs 36230

This theorem is used by:  mclsssvlem  36296  mclsax  36303  mclsind  36304  mclsppslem  36317