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Mathbox for Mario Carneiro |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mvtinf | Structured version Visualization version GIF version |
Description: Each variable typecode has infinitely many variables. (Contributed by Mario Carneiro, 18-Jul-2016.) |
Ref | Expression |
---|---|
mvtinf.f | β’ πΉ = (mVTβπ) |
mvtinf.y | β’ π = (mTypeβπ) |
Ref | Expression |
---|---|
mvtinf | β’ ((π β mFS β§ π β πΉ) β Β¬ (β‘π β {π}) β Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . . . 5 β’ (mCNβπ) = (mCNβπ) | |
2 | eqid 2733 | . . . . 5 β’ (mVRβπ) = (mVRβπ) | |
3 | mvtinf.y | . . . . 5 β’ π = (mTypeβπ) | |
4 | mvtinf.f | . . . . 5 β’ πΉ = (mVTβπ) | |
5 | eqid 2733 | . . . . 5 β’ (mTCβπ) = (mTCβπ) | |
6 | eqid 2733 | . . . . 5 β’ (mAxβπ) = (mAxβπ) | |
7 | eqid 2733 | . . . . 5 β’ (mStatβπ) = (mStatβπ) | |
8 | 1, 2, 3, 4, 5, 6, 7 | ismfs 34540 | . . . 4 β’ (π β mFS β (π β mFS β ((((mCNβπ) β© (mVRβπ)) = β β§ π:(mVRβπ)βΆ(mTCβπ)) β§ ((mAxβπ) β (mStatβπ) β§ βπ£ β πΉ Β¬ (β‘π β {π£}) β Fin)))) |
9 | 8 | ibi 267 | . . 3 β’ (π β mFS β ((((mCNβπ) β© (mVRβπ)) = β β§ π:(mVRβπ)βΆ(mTCβπ)) β§ ((mAxβπ) β (mStatβπ) β§ βπ£ β πΉ Β¬ (β‘π β {π£}) β Fin))) |
10 | 9 | simprrd 773 | . 2 β’ (π β mFS β βπ£ β πΉ Β¬ (β‘π β {π£}) β Fin) |
11 | sneq 4639 | . . . . . 6 β’ (π£ = π β {π£} = {π}) | |
12 | 11 | imaeq2d 6060 | . . . . 5 β’ (π£ = π β (β‘π β {π£}) = (β‘π β {π})) |
13 | 12 | eleq1d 2819 | . . . 4 β’ (π£ = π β ((β‘π β {π£}) β Fin β (β‘π β {π}) β Fin)) |
14 | 13 | notbid 318 | . . 3 β’ (π£ = π β (Β¬ (β‘π β {π£}) β Fin β Β¬ (β‘π β {π}) β Fin)) |
15 | 14 | rspccva 3612 | . 2 β’ ((βπ£ β πΉ Β¬ (β‘π β {π£}) β Fin β§ π β πΉ) β Β¬ (β‘π β {π}) β Fin) |
16 | 10, 15 | sylan 581 | 1 β’ ((π β mFS β§ π β πΉ) β Β¬ (β‘π β {π}) β Fin) |
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-xp 5683 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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