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Mirrors > Home > MPE Home > Th. List > Mathboxes > maxsta | Structured version Visualization version GIF version |
Description: An axiom is a statement. (Contributed by Mario Carneiro, 18-Jul-2016.) |
Ref | Expression |
---|---|
maxsta.a | ⊢ 𝐴 = (mAx‘𝑇) |
maxsta.s | ⊢ 𝑆 = (mStat‘𝑇) |
Ref | Expression |
---|---|
maxsta | ⊢ (𝑇 ∈ mFS → 𝐴 ⊆ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2735 | . . . 4 ⊢ (mCN‘𝑇) = (mCN‘𝑇) | |
2 | eqid 2735 | . . . 4 ⊢ (mVR‘𝑇) = (mVR‘𝑇) | |
3 | eqid 2735 | . . . 4 ⊢ (mType‘𝑇) = (mType‘𝑇) | |
4 | eqid 2735 | . . . 4 ⊢ (mVT‘𝑇) = (mVT‘𝑇) | |
5 | eqid 2735 | . . . 4 ⊢ (mTC‘𝑇) = (mTC‘𝑇) | |
6 | maxsta.a | . . . 4 ⊢ 𝐴 = (mAx‘𝑇) | |
7 | maxsta.s | . . . 4 ⊢ 𝑆 = (mStat‘𝑇) | |
8 | 1, 2, 3, 4, 5, 6, 7 | ismfs 35534 | . . 3 ⊢ (𝑇 ∈ mFS → (𝑇 ∈ mFS ↔ ((((mCN‘𝑇) ∩ (mVR‘𝑇)) = ∅ ∧ (mType‘𝑇):(mVR‘𝑇)⟶(mTC‘𝑇)) ∧ (𝐴 ⊆ 𝑆 ∧ ∀𝑣 ∈ (mVT‘𝑇) ¬ (◡(mType‘𝑇) “ {𝑣}) ∈ Fin)))) |
9 | 8 | ibi 267 | . 2 ⊢ (𝑇 ∈ mFS → ((((mCN‘𝑇) ∩ (mVR‘𝑇)) = ∅ ∧ (mType‘𝑇):(mVR‘𝑇)⟶(mTC‘𝑇)) ∧ (𝐴 ⊆ 𝑆 ∧ ∀𝑣 ∈ (mVT‘𝑇) ¬ (◡(mType‘𝑇) “ {𝑣}) ∈ Fin))) |
10 | 9 | simprld 772 | 1 ⊢ (𝑇 ∈ mFS → 𝐴 ⊆ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ∩ cin 3962 ⊆ wss 3963 ∅c0 4339 {csn 4631 ◡ccnv 5688 “ cima 5692 ⟶wf 6559 ‘cfv 6563 Fincfn 8984 mCNcmcn 35445 mVRcmvar 35446 mTypecmty 35447 mVTcmvt 35448 mTCcmtc 35449 mAxcmax 35450 mStatcmsta 35460 mFScmfs 35461 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-mfs 35481 |
This theorem is referenced by: mclsssvlem 35547 mclsax 35554 mclsind 35555 mclsppslem 35568 |
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