Metamath Proof Explorer

Theorem hilbert1.1

Description: There is a line through any two distinct points. Hilbert's axiom I.1 for geometry. (Contributed by Scott Fenton, 29-Oct-2013) (Revised by Mario Carneiro, 19-Apr-2014)

Ref Expression
Assertion hilbert1.1 
|- ( ( N e. NN /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ P =/= Q ) ) -> E. x e. LinesEE ( P e. x /\ Q e. x ) )

Proof

Step Hyp Ref Expression
1 simp1 
 |-  ( ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ P =/= Q ) -> P e. ( EE ` N ) )
2 simp2 
 |-  ( ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ P =/= Q ) -> Q e. ( EE ` N ) )
3 simp3 
 |-  ( ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ P =/= Q ) -> P =/= Q )
4 eqidd 
 |-  ( ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ P =/= Q ) -> ( P Line Q ) = ( P Line Q ) )
5 neeq1 
 |-  ( p = P -> ( p =/= q <-> P =/= q ) )
6 oveq1 
 |-  ( p = P -> ( p Line q ) = ( P Line q ) )
7 6 eqeq2d 
 |-  ( p = P -> ( ( P Line Q ) = ( p Line q ) <-> ( P Line Q ) = ( P Line q ) ) )
8 5 7 anbi12d 
 |-  ( p = P -> ( ( p =/= q /\ ( P Line Q ) = ( p Line q ) ) <-> ( P =/= q /\ ( P Line Q ) = ( P Line q ) ) ) )
9 neeq2 
 |-  ( q = Q -> ( P =/= q <-> P =/= Q ) )
10 oveq2 
 |-  ( q = Q -> ( P Line q ) = ( P Line Q ) )
11 10 eqeq2d 
 |-  ( q = Q -> ( ( P Line Q ) = ( P Line q ) <-> ( P Line Q ) = ( P Line Q ) ) )
12 9 11 anbi12d 
 |-  ( q = Q -> ( ( P =/= q /\ ( P Line Q ) = ( P Line q ) ) <-> ( P =/= Q /\ ( P Line Q ) = ( P Line Q ) ) ) )
13 8 12 rspc2ev 
 |-  ( ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ ( P =/= Q /\ ( P Line Q ) = ( P Line Q ) ) ) -> E. p e. ( EE ` N ) E. q e. ( EE ` N ) ( p =/= q /\ ( P Line Q ) = ( p Line q ) ) )
14 1 2 3 4 13 syl112anc 
 |-  ( ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ P =/= Q ) -> E. p e. ( EE ` N ) E. q e. ( EE ` N ) ( p =/= q /\ ( P Line Q ) = ( p Line q ) ) )
15 fveq2 
 |-  ( n = N -> ( EE ` n ) = ( EE ` N ) )
16 15 rexeqdv 
 |-  ( n = N -> ( E. q e. ( EE ` n ) ( p =/= q /\ ( P Line Q ) = ( p Line q ) ) <-> E. q e. ( EE ` N ) ( p =/= q /\ ( P Line Q ) = ( p Line q ) ) ) )
17 15 16 rexeqbidv 
 |-  ( n = N -> ( E. p e. ( EE ` n ) E. q e. ( EE ` n ) ( p =/= q /\ ( P Line Q ) = ( p Line q ) ) <-> E. p e. ( EE ` N ) E. q e. ( EE ` N ) ( p =/= q /\ ( P Line Q ) = ( p Line q ) ) ) )
18 17 rspcev 
 |-  ( ( N e. NN /\ E. p e. ( EE ` N ) E. q e. ( EE ` N ) ( p =/= q /\ ( P Line Q ) = ( p Line q ) ) ) -> E. n e. NN E. p e. ( EE ` n ) E. q e. ( EE ` n ) ( p =/= q /\ ( P Line Q ) = ( p Line q ) ) )
19 14 18 sylan2 
 |-  ( ( N e. NN /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ P =/= Q ) ) -> E. n e. NN E. p e. ( EE ` n ) E. q e. ( EE ` n ) ( p =/= q /\ ( P Line Q ) = ( p Line q ) ) )
20 ellines 
 |-  ( ( P Line Q ) e. LinesEE <-> E. n e. NN E. p e. ( EE ` n ) E. q e. ( EE ` n ) ( p =/= q /\ ( P Line Q ) = ( p Line q ) ) )
21 19 20 sylibr 
 |-  ( ( N e. NN /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ P =/= Q ) ) -> ( P Line Q ) e. LinesEE )
22 linerflx1 
 |-  ( ( N e. NN /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ P =/= Q ) ) -> P e. ( P Line Q ) )
23 linerflx2 
 |-  ( ( N e. NN /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ P =/= Q ) ) -> Q e. ( P Line Q ) )
24 eleq2 
 |-  ( x = ( P Line Q ) -> ( P e. x <-> P e. ( P Line Q ) ) )
25 eleq2 
 |-  ( x = ( P Line Q ) -> ( Q e. x <-> Q e. ( P Line Q ) ) )
26 24 25 anbi12d 
 |-  ( x = ( P Line Q ) -> ( ( P e. x /\ Q e. x ) <-> ( P e. ( P Line Q ) /\ Q e. ( P Line Q ) ) ) )
27 26 rspcev 
 |-  ( ( ( P Line Q ) e. LinesEE /\ ( P e. ( P Line Q ) /\ Q e. ( P Line Q ) ) ) -> E. x e. LinesEE ( P e. x /\ Q e. x ) )
28 21 22 23 27 syl12anc 
 |-  ( ( N e. NN /\ ( P e. ( EE ` N ) /\ Q e. ( EE ` N ) /\ P =/= Q ) ) -> E. x e. LinesEE ( P e. x /\ Q e. x ) )