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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 2738 | . . . 4 ⊢ (mCN‘𝑇) = (mCN‘𝑇) | |
2 | eqid 2738 | . . . 4 ⊢ (mVR‘𝑇) = (mVR‘𝑇) | |
3 | eqid 2738 | . . . 4 ⊢ (mType‘𝑇) = (mType‘𝑇) | |
4 | eqid 2738 | . . . 4 ⊢ (mVT‘𝑇) = (mVT‘𝑇) | |
5 | eqid 2738 | . . . 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 33947 | . . 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 771 | 1 ⊢ (𝑇 ∈ mFS → 𝐴 ⊆ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3063 ∩ cin 3908 ⊆ wss 3909 ∅c0 4281 {csn 4585 ◡ccnv 5631 “ cima 5635 ⟶wf 6490 ‘cfv 6494 Fincfn 8842 mCNcmcn 33858 mVRcmvar 33859 mTypecmty 33860 mVTcmvt 33861 mTCcmtc 33862 mAxcmax 33863 mStatcmsta 33873 mFScmfs 33874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3064 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-opab 5167 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-fv 6502 df-mfs 33894 |
This theorem is referenced by: mclsssvlem 33960 mclsax 33967 mclsind 33968 mclsppslem 33981 |
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