Theorem basis1 12214

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 12211	. . . 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 3270	. . . . 5 |- ( x = C -> ( x i^i y ) = ( C i^i y ) )
4	3	pweqd 3515	. . . . . . 7 |- ( x = C -> ~P ( x i^i y ) = ~P ( C i^i y ) )
5	4	ineq2d 3277	. . . . . 6 |- ( x = C -> ( B i^i ~P ( x i^i y ) ) = ( B i^i ~P ( C i^i y ) ) )
6	5	unieqd 3747	. . . . 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 3128	. . . 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 3271	. . . . 5 |- ( y = D -> ( C i^i y ) = ( C i^i D ) )
9	8	pweqd 3515	. . . . . . 7 |- ( y = D -> ~P ( C i^i y ) = ~P ( C i^i D ) )
10	9	ineq2d 3277	. . . . . 6 |- ( y = D -> ( B i^i ~P ( C i^i y ) ) = ( B i^i ~P ( C i^i D ) ) )
11	10	unieqd 3747	. . . . 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 3128	. . . 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 2802	. . 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 1179	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 962   = wceq 1331   e. wcel 1480   A.wral 2416   i^i cin 3070   C_ wss 3071   ~Pcpw 3510   U.cuni 3736   TopBasesctb 12209

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-in 3077  df-ss 3084  df-pw 3512  df-uni 3737  df-bases 12210

This theorem is referenced by: (None)