Theorem basis1 12839

Description: Property of a basis. (Contributed by NM, 16-Jul-2006.)

Assertion

Ref	Expression
basis1	|- ( ( B e. TopBases /\ C e. B /\ D e. B ) -> ( C i^i D ) C_ U. ( B i^i ~P ( C i^i D ) ) )

Proof of Theorem basis1

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isbasisg 12836	. . . 4 |- ( B e. TopBases -> ( B e. TopBases <-> A. x e. B A. y e. B ( x i^i y ) C_ U. ( B i^i ~P ( x i^i y ) ) ) )
2	1	ibi 175	. . 3 |- ( B e. TopBases -> A. x e. B A. y e. B ( x i^i y ) C_ U. ( B i^i ~P ( x i^i y ) ) )
3		ineq1 3321	. . . . 5 |- ( x = C -> ( x i^i y ) = ( C i^i y ) )
4	3	pweqd 3571	. . . . . . 7 |- ( x = C -> ~P ( x i^i y ) = ~P ( C i^i y ) )
5	4	ineq2d 3328	. . . . . 6 |- ( x = C -> ( B i^i ~P ( x i^i y ) ) = ( B i^i ~P ( C i^i y ) ) )
6	5	unieqd 3807	. . . . 5 |- ( x = C -> U. ( B i^i ~P ( x i^i y ) ) = U. ( B i^i ~P ( C i^i y ) ) )
7	3, 6	sseq12d 3178	. . . 4 |- ( x = C -> ( ( x i^i y ) C_ U. ( B i^i ~P ( x i^i y ) ) <-> ( C i^i y ) C_ U. ( B i^i ~P ( C i^i y ) ) ) )
8		ineq2 3322	. . . . 5 |- ( y = D -> ( C i^i y ) = ( C i^i D ) )
9	8	pweqd 3571	. . . . . . 7 |- ( y = D -> ~P ( C i^i y ) = ~P ( C i^i D ) )
10	9	ineq2d 3328	. . . . . 6 |- ( y = D -> ( B i^i ~P ( C i^i y ) ) = ( B i^i ~P ( C i^i D ) ) )
11	10	unieqd 3807	. . . . 5 |- ( y = D -> U. ( B i^i ~P ( C i^i y ) ) = U. ( B i^i ~P ( C i^i D ) ) )
12	8, 11	sseq12d 3178	. . . 4 |- ( y = D -> ( ( C i^i y ) C_ U. ( B i^i ~P ( C i^i y ) ) <-> ( C i^i D ) C_ U. ( B i^i ~P ( C i^i D ) ) ) )
13	7, 12	rspc2v 2847	. . 3 |- ( ( C e. B /\ D e. B ) -> ( A. x e. B A. y e. B ( x i^i y ) C_ U. ( B i^i ~P ( x i^i y ) ) -> ( C i^i D ) C_ U. ( B i^i ~P ( C i^i D ) ) ) )
14	2, 13	syl5com 29	. 2 |- ( B e. TopBases -> ( ( C e. B /\ D e. B ) -> ( C i^i D ) C_ U. ( B i^i ~P ( C i^i D ) ) ) )
15	14	3impib 1196	1 |- ( ( B e. TopBases /\ C e. B /\ D e. B ) -> ( C i^i D ) C_ U. ( B i^i ~P ( C i^i D ) ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 973   = wceq 1348   e. wcel 2141   A.wral 2448   i^i cin 3120   C_ wss 3121   ~Pcpw 3566   U.cuni 3796   TopBasesctb 12834

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-in 3127  df-ss 3134  df-pw 3568  df-uni 3797  df-bases 12835

This theorem is referenced by: (None)