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Mirrors > Home > MPE Home > Th. List > Mathboxes > mpstrcl | Structured version Visualization version GIF version |
Description: The elements of a pre-statement are sets. (Contributed by Mario Carneiro, 18-Jul-2016.) |
Ref | Expression |
---|---|
mpstssv.p | β’ π = (mPreStβπ) |
Ref | Expression |
---|---|
mpstrcl | β’ (β¨π·, π», π΄β© β π β (π· β V β§ π» β V β§ π΄ β V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ot 4599 | . . 3 β’ β¨π·, π», π΄β© = β¨β¨π·, π»β©, π΄β© | |
2 | mpstssv.p | . . . . 5 β’ π = (mPreStβπ) | |
3 | 2 | mpstssv 34197 | . . . 4 β’ π β ((V Γ V) Γ V) |
4 | 3 | sseli 3944 | . . 3 β’ (β¨π·, π», π΄β© β π β β¨π·, π», π΄β© β ((V Γ V) Γ V)) |
5 | 1, 4 | eqeltrrid 2839 | . 2 β’ (β¨π·, π», π΄β© β π β β¨β¨π·, π»β©, π΄β© β ((V Γ V) Γ V)) |
6 | opelxp 5673 | . . . 4 β’ (β¨π·, π»β© β (V Γ V) β (π· β V β§ π» β V)) | |
7 | 6 | anbi1i 625 | . . 3 β’ ((β¨π·, π»β© β (V Γ V) β§ π΄ β V) β ((π· β V β§ π» β V) β§ π΄ β V)) |
8 | opelxp 5673 | . . 3 β’ (β¨β¨π·, π»β©, π΄β© β ((V Γ V) Γ V) β (β¨π·, π»β© β (V Γ V) β§ π΄ β V)) | |
9 | df-3an 1090 | . . 3 β’ ((π· β V β§ π» β V β§ π΄ β V) β ((π· β V β§ π» β V) β§ π΄ β V)) | |
10 | 7, 8, 9 | 3bitr4i 303 | . 2 β’ (β¨β¨π·, π»β©, π΄β© β ((V Γ V) Γ V) β (π· β V β§ π» β V β§ π΄ β V)) |
11 | 5, 10 | sylib 217 | 1 β’ (β¨π·, π», π΄β© β π β (π· β V β§ π» β V β§ π΄ β V)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 Vcvv 3447 β¨cop 4596 β¨cotp 4598 Γ cxp 5635 βcfv 6500 mPreStcmpst 34131 |
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-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
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-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-ot 4599 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-iota 6452 df-fun 6502 df-fv 6508 df-mpst 34151 |
This theorem is referenced by: elmsta 34206 mclsax 34227 |
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