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Theorem maxsta 30705
 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 2610 . . . . 5 (mCN‘𝑇) = (mCN‘𝑇)
2 eqid 2610 . . . . 5 (mVR‘𝑇) = (mVR‘𝑇)
3 eqid 2610 . . . . 5 (mType‘𝑇) = (mType‘𝑇)
4 eqid 2610 . . . . 5 (mVT‘𝑇) = (mVT‘𝑇)
5 eqid 2610 . . . . 5 (mTC‘𝑇) = (mTC‘𝑇)
6 maxsta.a . . . . 5 𝐴 = (mAx‘𝑇)
7 maxsta.s . . . . 5 𝑆 = (mStat‘𝑇)
81, 2, 3, 4, 5, 6, 7ismfs 30700 . . . 4 (𝑇 ∈ mFS → (𝑇 ∈ mFS ↔ ((((mCN‘𝑇) ∩ (mVR‘𝑇)) = ∅ ∧ (mType‘𝑇):(mVR‘𝑇)⟶(mTC‘𝑇)) ∧ (𝐴𝑆 ∧ ∀𝑣 ∈ (mVT‘𝑇) ¬ ((mType‘𝑇) “ {𝑣}) ∈ Fin))))
98ibi 255 . . 3 (𝑇 ∈ mFS → ((((mCN‘𝑇) ∩ (mVR‘𝑇)) = ∅ ∧ (mType‘𝑇):(mVR‘𝑇)⟶(mTC‘𝑇)) ∧ (𝐴𝑆 ∧ ∀𝑣 ∈ (mVT‘𝑇) ¬ ((mType‘𝑇) “ {𝑣}) ∈ Fin)))
109simprd 478 . 2 (𝑇 ∈ mFS → (𝐴𝑆 ∧ ∀𝑣 ∈ (mVT‘𝑇) ¬ ((mType‘𝑇) “ {𝑣}) ∈ Fin))
1110simpld 474 1 (𝑇 ∈ mFS → 𝐴𝑆)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  {csn 4125  ◡ccnv 5037   “ cima 5041  ⟶wf 5800  ‘cfv 5804  Fincfn 7841  mCNcmcn 30611  mVRcmvar 30612  mTypecmty 30613  mVTcmvt 30614  mTCcmtc 30615  mAxcmax 30616  mStatcmsta 30626  mFScmfs 30627 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-mfs 30647 This theorem is referenced by:  mclsssvlem  30713  mclsax  30720  mclsind  30721  mclsppslem  30734
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