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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 2823 | . . . 4 ⊢ (mType‘𝑇) = (mType‘𝑇) | |
4 | eqid 2823 | . . . 4 ⊢ (mVT‘𝑇) = (mVT‘𝑇) | |
5 | eqid 2823 | . . . 4 ⊢ (mTC‘𝑇) = (mTC‘𝑇) | |
6 | eqid 2823 | . . . 4 ⊢ (mAx‘𝑇) = (mAx‘𝑇) | |
7 | eqid 2823 | . . . 4 ⊢ (mStat‘𝑇) = (mStat‘𝑇) | |
8 | 1, 2, 3, 4, 5, 6, 7 | ismfs 32798 | . . 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 269 | . 2 ⊢ (𝑇 ∈ mFS → (((𝐶 ∩ 𝑉) = ∅ ∧ (mType‘𝑇):𝑉⟶(mTC‘𝑇)) ∧ ((mAx‘𝑇) ⊆ (mStat‘𝑇) ∧ ∀𝑣 ∈ (mVT‘𝑇) ¬ (◡(mType‘𝑇) “ {𝑣}) ∈ Fin))) |
10 | 9 | simplld 766 | 1 ⊢ (𝑇 ∈ mFS → (𝐶 ∩ 𝑉) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ∩ cin 3937 ⊆ wss 3938 ∅c0 4293 {csn 4569 ◡ccnv 5556 “ cima 5560 ⟶wf 6353 ‘cfv 6357 Fincfn 8511 mCNcmcn 32709 mVRcmvar 32710 mTypecmty 32711 mVTcmvt 32712 mTCcmtc 32713 mAxcmax 32714 mStatcmsta 32724 mFScmfs 32725 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-mfs 32745 |
This theorem is referenced by: (None) |
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