Theorem islln 34792

Description: The predicate "is a lattice line". (Contributed by NM, 16-Jun-2012.)

Hypotheses

Ref	Expression
llnset.b	|- B = ( Base ` K )
llnset.c	|- C = ( <o ` K )
llnset.a	|- A = ( Atoms ` K )
llnset.n	|- N = ( LLines ` K )

Assertion

Ref	Expression
islln	|- ( K e. D -> ( X e. N <-> ( X e. B /\ E. p e. A p C X ) ) )

Distinct variable groups:   A, p   K, p   X, p

Allowed substitution hints:   B( p)   C( p)   D( p)   N( p)

Proof of Theorem islln

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		llnset.b	. . . 4 |- B = ( Base ` K )
2		llnset.c	. . . 4 |- C = ( <o ` K )
3		llnset.a	. . . 4 |- A = ( Atoms ` K )
4		llnset.n	. . . 4 |- N = ( LLines ` K )
5	1, 2, 3, 4	llnset 34791	. . 3 |- ( K e. D -> N = { x e. B | E. p e. A p C x } )
6	5	eleq2d 2687	. 2 |- ( K e. D -> ( X e. N <-> X e. { x e. B | E. p e. A p C x } ) )
7		breq2 4657	. . . 4 |- ( x = X -> ( p C x <-> p C X ) )
8	7	rexbidv 3052	. . 3 |- ( x = X -> ( E. p e. A p C x <-> E. p e. A p C X ) )
9	8	elrab 3363	. 2 |- ( X e. { x e. B | E. p e. A p C x } <-> ( X e. B /\ E. p e. A p C X ) )
10	6, 9	syl6bb 276	1 |- ( K e. D -> ( X e. N <-> ( X e. B /\ E. p e. A p C X ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   {crab 2916   class class class wbr 4653   ` cfv 5888   Basecbs 15857   <o ccvr 34549   Atomscatm 34550   LLinesclln 34777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-llines 34784

This theorem is referenced by:  islln4  34793  llni  34794  llnbase  34795  llnnleat  34799