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Theorem maxsta 33952
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 → 𝐴𝑆)

Proof of Theorem maxsta
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 eqid 2738 . . . 4 (mCN‘𝑇) = (mCN‘𝑇)
2 eqid 2738 . . . 4 (mVR‘𝑇) = (mVR‘𝑇)
3 eqid 2738 . . . 4 (mType‘𝑇) = (mType‘𝑇)
4 eqid 2738 . . . 4 (mVT‘𝑇) = (mVT‘𝑇)
5 eqid 2738 . . . 4 (mTC‘𝑇) = (mTC‘𝑇)
6 maxsta.a . . . 4 𝐴 = (mAx‘𝑇)
7 maxsta.s . . . 4 𝑆 = (mStat‘𝑇)
81, 2, 3, 4, 5, 6, 7ismfs 33947 . . 3 (𝑇 ∈ mFS → (𝑇 ∈ mFS ↔ ((((mCN‘𝑇) ∩ (mVR‘𝑇)) = ∅ ∧ (mType‘𝑇):(mVR‘𝑇)⟶(mTC‘𝑇)) ∧ (𝐴𝑆 ∧ ∀𝑣 ∈ (mVT‘𝑇) ¬ ((mType‘𝑇) “ {𝑣}) ∈ Fin))))
98ibi 267 . 2 (𝑇 ∈ mFS → ((((mCN‘𝑇) ∩ (mVR‘𝑇)) = ∅ ∧ (mType‘𝑇):(mVR‘𝑇)⟶(mTC‘𝑇)) ∧ (𝐴𝑆 ∧ ∀𝑣 ∈ (mVT‘𝑇) ¬ ((mType‘𝑇) “ {𝑣}) ∈ Fin)))
109simprld 771 1 (𝑇 ∈ mFS → 𝐴𝑆)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3063  cin 3908  wss 3909  c0 4281  {csn 4585  ccnv 5631  cima 5635  wf 6490  cfv 6494  Fincfn 8842  mCNcmcn 33858  mVRcmvar 33859  mTypecmty 33860  mVTcmvt 33861  mTCcmtc 33862  mAxcmax 33863  mStatcmsta 33873  mFScmfs 33874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3064  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-opab 5167  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6446  df-fun 6496  df-fn 6497  df-f 6498  df-fv 6502  df-mfs 33894
This theorem is referenced by:  mclsssvlem  33960  mclsax  33967  mclsind  33968  mclsppslem  33981
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