Theorem tglnne0 25535

Description: A line A has at least one point. (Contributed by Thierry Arnoux, 4-Mar-2020.)

Hypotheses

Ref	Expression
tglnne0.l	|- L = (LineG ` G )
tglnne0.g	|- ( ph -> G e. TarskiG )
tglnne0.1	|- ( ph -> A e. ran L )

Assertion

Ref	Expression
tglnne0	|- ( ph -> A =/= (/) )

Proof of Theorem tglnne0

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . 5 |- ( Base ` G ) = ( Base ` G )
2		eqid 2622	. . . . 5 |- (Itv ` G ) = (Itv ` G )
3		tglnne0.l	. . . . 5 |- L = (LineG ` G )
4		tglnne0.g	. . . . . 6 |- ( ph -> G e. TarskiG )
5	4	ad3antrrr 766	. . . . 5 |- ( ( ( ( ph /\ x e. ( Base ` G ) ) /\ y e. ( Base ` G ) ) /\ ( A = ( x L y ) /\ x =/= y ) ) -> G e. TarskiG )
6		simpllr 799	. . . . 5 |- ( ( ( ( ph /\ x e. ( Base ` G ) ) /\ y e. ( Base ` G ) ) /\ ( A = ( x L y ) /\ x =/= y ) ) -> x e. ( Base ` G ) )
7		simplr 792	. . . . 5 |- ( ( ( ( ph /\ x e. ( Base ` G ) ) /\ y e. ( Base ` G ) ) /\ ( A = ( x L y ) /\ x =/= y ) ) -> y e. ( Base ` G ) )
8		simprr 796	. . . . 5 |- ( ( ( ( ph /\ x e. ( Base ` G ) ) /\ y e. ( Base ` G ) ) /\ ( A = ( x L y ) /\ x =/= y ) ) -> x =/= y )
9	1, 2, 3, 5, 6, 7, 8	tglinerflx1 25528	. . . 4 |- ( ( ( ( ph /\ x e. ( Base ` G ) ) /\ y e. ( Base ` G ) ) /\ ( A = ( x L y ) /\ x =/= y ) ) -> x e. ( x L y ) )
10		simprl 794	. . . 4 |- ( ( ( ( ph /\ x e. ( Base ` G ) ) /\ y e. ( Base ` G ) ) /\ ( A = ( x L y ) /\ x =/= y ) ) -> A = ( x L y ) )
11	9, 10	eleqtrrd 2704	. . 3 |- ( ( ( ( ph /\ x e. ( Base ` G ) ) /\ y e. ( Base ` G ) ) /\ ( A = ( x L y ) /\ x =/= y ) ) -> x e. A )
12		ne0i 3921	. . 3 |- ( x e. A -> A =/= (/) )
13	11, 12	syl 17	. 2 |- ( ( ( ( ph /\ x e. ( Base ` G ) ) /\ y e. ( Base ` G ) ) /\ ( A = ( x L y ) /\ x =/= y ) ) -> A =/= (/) )
14		tglnne0.1	. . 3 |- ( ph -> A e. ran L )
15	1, 2, 3, 4, 14	tgisline 25522	. 2 |- ( ph -> E. x e. ( Base ` G ) E. y e. ( Base ` G ) ( A = ( x L y ) /\ x =/= y ) )
16	13, 15	r19.29vva 3081	1 |- ( ph -> A =/= (/) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   (/)c0 3915   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-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-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-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-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-trkgc 25347  df-trkgb 25348  df-trkgcb 25349  df-trkg 25352

This theorem is referenced by:  hpgerlem  25657