Theorem ispointN 35028

Description: The predicate "is a point". (Contributed by NM, 2-Oct-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
ispoint.a	|- A = ( Atoms ` K )
ispoint.p	|- P = ( Points ` K )

Assertion

Ref	Expression
ispointN	|- ( K e. D -> ( X e. P <-> E. a e. A X = { a } ) )

Distinct variable groups:   A, a   X, a

Allowed substitution hints:   D( a)   P( a)   K( a)

Proof of Theorem ispointN

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ispoint.a	. . . 4 |- A = ( Atoms ` K )
2		ispoint.p	. . . 4 |- P = ( Points ` K )
3	1, 2	pointsetN 35027	. . 3 |- ( K e. D -> P = { x | E. a e. A x = { a } } )
4	3	eleq2d 2687	. 2 |- ( K e. D -> ( X e. P <-> X e. { x | E. a e. A x = { a } } ) )
5		snex 4908	. . . . 5 |- { a } e. _V
6		eleq1 2689	. . . . 5 |- ( X = { a } -> ( X e. _V <-> { a } e. _V ) )
7	5, 6	mpbiri 248	. . . 4 |- ( X = { a } -> X e. _V )
8	7	rexlimivw 3029	. . 3 |- ( E. a e. A X = { a } -> X e. _V )
9		eqeq1 2626	. . . 4 |- ( x = X -> ( x = { a } <-> X = { a } ) )
10	9	rexbidv 3052	. . 3 |- ( x = X -> ( E. a e. A x = { a } <-> E. a e. A X = { a } ) )
11	8, 10	elab3 3358	. 2 |- ( X e. { x | E. a e. A x = { a } } <-> E. a e. A X = { a } )
12	4, 11	syl6bb 276	1 |- ( K e. D -> ( X e. P <-> E. a e. A X = { a } ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913   _Vcvv 3200   {csn 4177   ` cfv 5888   Atomscatm 34550   PointscpointsN 34781

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

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-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  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-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-pointsN 34788

This theorem is referenced by:  atpointN  35029  pointpsubN  35037