Theorem isline 17220

Description: The predicate "is a line".

Hypotheses

Ref	Expression
isline.l	|- L = (le` K)
isline.j	|- J = (join` K)
isline.a	|- A = (AtomsNEW` K)
isline.n	|- N = (Lines` K)

Assertion

Ref	Expression
isline	|- (K e. D -> (X e. N <-> E.q e. A E.r e. A (q =/= r /\ X = {p e. A | pL(qJr)})))

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

Proof of Theorem isline

Step	Hyp	Ref	Expression
1		isline.l	. . . 4 |- L = (le` K)
2		isline.j	. . . 4 |- J = (join` K)
3		isline.a	. . . 4 |- A = (AtomsNEW` K)
4		isline.n	. . . 4 |- N = (Lines` K)
5	1, 2, 3, 4	lineset 17219	. . 3 |- (K e. D -> N = {x | E.q e. A E.r e. A (q =/= r /\ x = {p e. A | pL(qJr)})})
6	5	eleq2d 1964	. 2 |- (K e. D -> (X e. N <-> X e. {x | E.q e. A E.r e. A (q =/= r /\ x = {p e. A | pL(qJr)})}))
7		fvex 4689	. . . . . . . . 9 |- (AtomsNEW` K) e. _V
8	3, 7	eqeltri 1967	. . . . . . . 8 |- A e. _V
9	8	rabex 3461	. . . . . . 7 |- {p e. A | pL(qJr)} e. _V
10		eleq1 1957	. . . . . . 7 |- (X = {p e. A | pL(qJr)} -> (X e. _V <-> {p e. A | pL(qJr)} e. _V))
11	9, 10	mpbiri 211	. . . . . 6 |- (X = {p e. A | pL(qJr)} -> X e. _V)
12	11	adantl 424	. . . . 5 |- ((q =/= r /\ X = {p e. A | pL(qJr)}) -> X e. _V)
13	12	a1i 8	. . . 4 |- ((q e. A /\ r e. A) -> ((q =/= r /\ X = {p e. A | pL(qJr)}) -> X e. _V))
14	13	r19.23aivv 2217	. . 3 |- (E.q e. A E.r e. A (q =/= r /\ X = {p e. A | pL(qJr)}) -> X e. _V)
15		eqeq1 1890	. . . . 5 |- (x = X -> (x = {p e. A | pL(qJr)} <-> X = {p e. A | pL(qJr)}))
16	15	anbi2d 678	. . . 4 |- (x = X -> ((q =/= r /\ x = {p e. A | pL(qJr)}) <-> (q =/= r /\ X = {p e. A | pL(qJr)})))
17	16	2rexbidv 2141	. . 3 |- (x = X -> (E.q e. A E.r e. A (q =/= r /\ x = {p e. A | pL(qJr)}) <-> E.q e. A E.r e. A (q =/= r /\ X = {p e. A | pL(qJr)})))
18	14, 17	elab3 2412	. 2 |- (X e. {x | E.q e. A E.r e. A (q =/= r /\ x = {p e. A | pL(qJr)})} <-> E.q e. A E.r e. A (q =/= r /\ X = {p e. A | pL(qJr)}))
19	6, 18	syl6bb 595	1 |- (K e. D -> (X e. N <-> E.q e. A E.r e. A (q =/= r /\ X = {p e. A | pL(qJr)})))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  {cab 1871   =/= wne 2017  E.wrex 2106  {crab 2108  _Vcvv 2292   class class class wbr 3338  ` cfv 3998  (class class class)co 4884  lecple 16759  joincjn 16766  AtomsNEWcatm 16981  Linesclines 17211

This theorem is referenced by:  islinei 17221  linepsub 17232  isline2 17248

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-iun 3257  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fv 4014  df-opr 4886  df-mpt 5006  df-lines 17215

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