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| Mirrors > Home > MPE Home > Th. List > Mathboxes > mfsdisj | Structured version Visualization version GIF version | ||
| Description: The constants and variables of a formal system are disjoint. (Contributed by Mario Carneiro, 18-Jul-2016.) |
| Ref | Expression |
|---|---|
| mfsdisj.c | ⊢ 𝐶 = (mCN‘𝑇) |
| mfsdisj.v | ⊢ 𝑉 = (mVR‘𝑇) |
| Ref | Expression |
|---|---|
| mfsdisj | ⊢ (𝑇 ∈ mFS → (𝐶 ∩ 𝑉) = ∅) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mfsdisj.c | . . . 4 ⊢ 𝐶 = (mCN‘𝑇) | |
| 2 | mfsdisj.v | . . . 4 ⊢ 𝑉 = (mVR‘𝑇) | |
| 3 | eqid 2737 | . . . 4 ⊢ (mType‘𝑇) = (mType‘𝑇) | |
| 4 | eqid 2737 | . . . 4 ⊢ (mVT‘𝑇) = (mVT‘𝑇) | |
| 5 | eqid 2737 | . . . 4 ⊢ (mTC‘𝑇) = (mTC‘𝑇) | |
| 6 | eqid 2737 | . . . 4 ⊢ (mAx‘𝑇) = (mAx‘𝑇) | |
| 7 | eqid 2737 | . . . 4 ⊢ (mStat‘𝑇) = (mStat‘𝑇) | |
| 8 | 1, 2, 3, 4, 5, 6, 7 | ismfs 35554 | . . 3 ⊢ (𝑇 ∈ mFS → (𝑇 ∈ mFS ↔ (((𝐶 ∩ 𝑉) = ∅ ∧ (mType‘𝑇):𝑉⟶(mTC‘𝑇)) ∧ ((mAx‘𝑇) ⊆ (mStat‘𝑇) ∧ ∀𝑣 ∈ (mVT‘𝑇) ¬ (◡(mType‘𝑇) “ {𝑣}) ∈ Fin)))) |
| 9 | 8 | ibi 267 | . 2 ⊢ (𝑇 ∈ mFS → (((𝐶 ∩ 𝑉) = ∅ ∧ (mType‘𝑇):𝑉⟶(mTC‘𝑇)) ∧ ((mAx‘𝑇) ⊆ (mStat‘𝑇) ∧ ∀𝑣 ∈ (mVT‘𝑇) ¬ (◡(mType‘𝑇) “ {𝑣}) ∈ Fin))) |
| 10 | 9 | simplld 768 | 1 ⊢ (𝑇 ∈ mFS → (𝐶 ∩ 𝑉) = ∅) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 {csn 4626 ◡ccnv 5684 “ cima 5688 ⟶wf 6557 ‘cfv 6561 Fincfn 8985 mCNcmcn 35465 mVRcmvar 35466 mTypecmty 35467 mVTcmvt 35468 mTCcmtc 35469 mAxcmax 35470 mStatcmsta 35480 mFScmfs 35481 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-mfs 35501 |
| This theorem is referenced by: (None) |
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