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Theorem maxsta 31577
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 2651 . . . 4 (mCN‘𝑇) = (mCN‘𝑇)
2 eqid 2651 . . . 4 (mVR‘𝑇) = (mVR‘𝑇)
3 eqid 2651 . . . 4 (mType‘𝑇) = (mType‘𝑇)
4 eqid 2651 . . . 4 (mVT‘𝑇) = (mVT‘𝑇)
5 eqid 2651 . . . 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 31572 . . 3 (𝑇 ∈ mFS → (𝑇 ∈ mFS ↔ ((((mCN‘𝑇) ∩ (mVR‘𝑇)) = ∅ ∧ (mType‘𝑇):(mVR‘𝑇)⟶(mTC‘𝑇)) ∧ (𝐴𝑆 ∧ ∀𝑣 ∈ (mVT‘𝑇) ¬ ((mType‘𝑇) “ {𝑣}) ∈ Fin))))
98ibi 256 . 2 (𝑇 ∈ mFS → ((((mCN‘𝑇) ∩ (mVR‘𝑇)) = ∅ ∧ (mType‘𝑇):(mVR‘𝑇)⟶(mTC‘𝑇)) ∧ (𝐴𝑆 ∧ ∀𝑣 ∈ (mVT‘𝑇) ¬ ((mType‘𝑇) “ {𝑣}) ∈ Fin)))
109simprld 810 1 (𝑇 ∈ mFS → 𝐴𝑆)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1523  wcel 2030  wral 2941  cin 3606  wss 3607  c0 3948  {csn 4210  ccnv 5142  cima 5146  wf 5922  cfv 5926  Fincfn 7997  mCNcmcn 31483  mVRcmvar 31484  mTypecmty 31485  mVTcmvt 31486  mTCcmtc 31487  mAxcmax 31488  mStatcmsta 31498  mFScmfs 31499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-mfs 31519
This theorem is referenced by:  mclsssvlem  31585  mclsax  31592  mclsind  31593  mclsppslem  31606
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