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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 | β’ πΆ = (mCNβπ) |
mfsdisj.v | β’ π = (mVRβπ) |
Ref | Expression |
---|---|
mfsdisj | β’ (π β mFS β (πΆ β© π) = β ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mfsdisj.c | . . . 4 β’ πΆ = (mCNβπ) | |
2 | mfsdisj.v | . . . 4 β’ π = (mVRβπ) | |
3 | eqid 2733 | . . . 4 β’ (mTypeβπ) = (mTypeβπ) | |
4 | eqid 2733 | . . . 4 β’ (mVTβπ) = (mVTβπ) | |
5 | eqid 2733 | . . . 4 β’ (mTCβπ) = (mTCβπ) | |
6 | eqid 2733 | . . . 4 β’ (mAxβπ) = (mAxβπ) | |
7 | eqid 2733 | . . . 4 β’ (mStatβπ) = (mStatβπ) | |
8 | 1, 2, 3, 4, 5, 6, 7 | ismfs 34540 | . . 3 β’ (π β mFS β (π β mFS β (((πΆ β© π) = β β§ (mTypeβπ):πβΆ(mTCβπ)) β§ ((mAxβπ) β (mStatβπ) β§ βπ£ β (mVTβπ) Β¬ (β‘(mTypeβπ) β {π£}) β Fin)))) |
9 | 8 | ibi 267 | . 2 β’ (π β mFS β (((πΆ β© π) = β β§ (mTypeβπ):πβΆ(mTCβπ)) β§ ((mAxβπ) β (mStatβπ) β§ βπ£ β (mVTβπ) Β¬ (β‘(mTypeβπ) β {π£}) β Fin))) |
10 | 9 | simplld 767 | 1 β’ (π β mFS β (πΆ β© π) = β ) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 βwral 3062 β© cin 3948 β wss 3949 β c0 4323 {csn 4629 β‘ccnv 5676 β cima 5680 βΆwf 6540 βcfv 6544 Fincfn 8939 mCNcmcn 34451 mVRcmvar 34452 mTypecmty 34453 mVTcmvt 34454 mTCcmtc 34455 mAxcmax 34456 mStatcmsta 34466 mFScmfs 34467 |
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 2704 |
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 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-fv 6552 df-mfs 34487 |
This theorem is referenced by: (None) |
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