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Theorem maxsta 32875
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 2822 . . . 4 (mCN‘𝑇) = (mCN‘𝑇)
2 eqid 2822 . . . 4 (mVR‘𝑇) = (mVR‘𝑇)
3 eqid 2822 . . . 4 (mType‘𝑇) = (mType‘𝑇)
4 eqid 2822 . . . 4 (mVT‘𝑇) = (mVT‘𝑇)
5 eqid 2822 . . . 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 32870 . . 3 (𝑇 ∈ mFS → (𝑇 ∈ mFS ↔ ((((mCN‘𝑇) ∩ (mVR‘𝑇)) = ∅ ∧ (mType‘𝑇):(mVR‘𝑇)⟶(mTC‘𝑇)) ∧ (𝐴𝑆 ∧ ∀𝑣 ∈ (mVT‘𝑇) ¬ ((mType‘𝑇) “ {𝑣}) ∈ Fin))))
98ibi 270 . 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 399   = wceq 1538  wcel 2114  wral 3130  cin 3907  wss 3908  c0 4265  {csn 4539  ccnv 5531  cima 5535  wf 6330  cfv 6334  Fincfn 8496  mCNcmcn 32781  mVRcmvar 32782  mTypecmty 32783  mVTcmvt 32784  mTCcmtc 32785  mAxcmax 32786  mStatcmsta 32796  mFScmfs 32797
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rab 3139  df-v 3471  df-un 3913  df-in 3915  df-ss 3925  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-fv 6342  df-mfs 32817
This theorem is referenced by:  mclsssvlem  32883  mclsax  32890  mclsind  32891  mclsppslem  32904
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