Theorem tghilberti1 25532

Description: There is a line through any two distinct points. Hilbert's axiom I.1 for geometry. (Contributed by Thierry Arnoux, 25-May-2019.)

Hypotheses

Ref	Expression
tglineelsb2.p	|- B = ( Base ` G )
tglineelsb2.i	|- I = (Itv ` G )
tglineelsb2.l	|- L = (LineG ` G )
tglineelsb2.g	|- ( ph -> G e. TarskiG )
tglineelsb2.1	|- ( ph -> P e. B )
tglineelsb2.2	|- ( ph -> Q e. B )
tglineelsb2.4	|- ( ph -> P =/= Q )

Assertion

Ref	Expression
tghilberti1	|- ( ph -> E. x e. ran L ( P e. x /\ Q e. x ) )

Distinct variable groups:   x, B   x, G   x, I   x, L   x, P   x, Q   ph, x

Proof of Theorem tghilberti1

Step	Hyp	Ref	Expression
1		tglineelsb2.p	. . 3 |- B = ( Base ` G )
2		tglineelsb2.i	. . 3 |- I = (Itv ` G )
3		tglineelsb2.l	. . 3 |- L = (LineG ` G )
4		tglineelsb2.g	. . 3 |- ( ph -> G e. TarskiG )
5		tglineelsb2.1	. . 3 |- ( ph -> P e. B )
6		tglineelsb2.2	. . 3 |- ( ph -> Q e. B )
7		tglineelsb2.4	. . 3 |- ( ph -> P =/= Q )
8	1, 2, 3, 4, 5, 6, 7	tgelrnln 25525	. 2 |- ( ph -> ( P L Q ) e. ran L )
9	1, 2, 3, 4, 5, 6, 7	tglinerflx1 25528	. 2 |- ( ph -> P e. ( P L Q ) )
10	1, 2, 3, 4, 5, 6, 7	tglinerflx2 25529	. 2 |- ( ph -> Q e. ( P L Q ) )
11		eleq2 2690	. . . 4 |- ( x = ( P L Q ) -> ( P e. x <-> P e. ( P L Q ) ) )
12		eleq2 2690	. . . 4 |- ( x = ( P L Q ) -> ( Q e. x <-> Q e. ( P L Q ) ) )
13	11, 12	anbi12d 747	. . 3 |- ( x = ( P L Q ) -> ( ( P e. x /\ Q e. x ) <-> ( P e. ( P L Q ) /\ Q e. ( P L Q ) ) ) )
14	13	rspcev 3309	. 2 |- ( ( ( P L Q ) e. ran L /\ ( P e. ( P L Q ) /\ Q e. ( P L Q ) ) ) -> E. x e. ran L ( P e. x /\ Q e. x ) )
15	8, 9, 10, 14	syl12anc 1324	1 |- ( ph -> E. x e. ran L ( P e. x /\ Q e. x ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   ran crn 5115   ` cfv 5888  (class class class)co 6650   Basecbs 15857  TarskiGcstrkg 25329  Itvcitv 25335  LineGclng 25336

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-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-pw 4160  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-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-trkgc 25347  df-trkgb 25348  df-trkgcb 25349  df-trkg 25352

This theorem is referenced by:  tglinethrueu  25534