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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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