Theorem tglnpt2 25536

Description: Find a second point on a line. (Contributed by Thierry Arnoux, 18-Oct-2019.)

Hypotheses

Ref	Expression
tglnpt2.p	|- P = ( Base ` G )
tglnpt2.i	|- I = (Itv ` G )
tglnpt2.l	|- L = (LineG ` G )
tglnpt2.g	|- ( ph -> G e. TarskiG )
tglnpt2.a	|- ( ph -> A e. ran L )
tglnpt2.x	|- ( ph -> X e. A )

Assertion

Ref	Expression
tglnpt2	|- ( ph -> E. y e. A X =/= y )

Distinct variable groups:   y, A   y, X

Allowed substitution hints:   ph( y)   P( y)   G( y)   I( y)   L( y)

Proof of Theorem tglnpt2

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

Step	Hyp	Ref	Expression
1		tglnpt2.p	. . . . . 6 |- P = ( Base ` G )
2		tglnpt2.i	. . . . . 6 |- I = (Itv ` G )
3		tglnpt2.l	. . . . . 6 |- L = (LineG ` G )
4		tglnpt2.g	. . . . . . 7 |- ( ph -> G e. TarskiG )
5	4	ad4antr 768	. . . . . 6 |- ( ( ( ( ( ph /\ x e. P ) /\ z e. P ) /\ ( A = ( x L z ) /\ x =/= z ) ) /\ X = x ) -> G e. TarskiG )
6		simp-4r 807	. . . . . 6 |- ( ( ( ( ( ph /\ x e. P ) /\ z e. P ) /\ ( A = ( x L z ) /\ x =/= z ) ) /\ X = x ) -> x e. P )
7		simpllr 799	. . . . . 6 |- ( ( ( ( ( ph /\ x e. P ) /\ z e. P ) /\ ( A = ( x L z ) /\ x =/= z ) ) /\ X = x ) -> z e. P )
8		simplrr 801	. . . . . 6 |- ( ( ( ( ( ph /\ x e. P ) /\ z e. P ) /\ ( A = ( x L z ) /\ x =/= z ) ) /\ X = x ) -> x =/= z )
9	1, 2, 3, 5, 6, 7, 8	tglinerflx2 25529	. . . . 5 |- ( ( ( ( ( ph /\ x e. P ) /\ z e. P ) /\ ( A = ( x L z ) /\ x =/= z ) ) /\ X = x ) -> z e. ( x L z ) )
10		simplrl 800	. . . . 5 |- ( ( ( ( ( ph /\ x e. P ) /\ z e. P ) /\ ( A = ( x L z ) /\ x =/= z ) ) /\ X = x ) -> A = ( x L z ) )
11	9, 10	eleqtrrd 2704	. . . 4 |- ( ( ( ( ( ph /\ x e. P ) /\ z e. P ) /\ ( A = ( x L z ) /\ x =/= z ) ) /\ X = x ) -> z e. A )
12		simpr 477	. . . . 5 |- ( ( ( ( ( ph /\ x e. P ) /\ z e. P ) /\ ( A = ( x L z ) /\ x =/= z ) ) /\ X = x ) -> X = x )
13	12, 8	eqnetrd 2861	. . . 4 |- ( ( ( ( ( ph /\ x e. P ) /\ z e. P ) /\ ( A = ( x L z ) /\ x =/= z ) ) /\ X = x ) -> X =/= z )
14		neeq2 2857	. . . . 5 |- ( y = z -> ( X =/= y <-> X =/= z ) )
15	14	rspcev 3309	. . . 4 |- ( ( z e. A /\ X =/= z ) -> E. y e. A X =/= y )
16	11, 13, 15	syl2anc 693	. . 3 |- ( ( ( ( ( ph /\ x e. P ) /\ z e. P ) /\ ( A = ( x L z ) /\ x =/= z ) ) /\ X = x ) -> E. y e. A X =/= y )
17	4	ad4antr 768	. . . . . 6 |- ( ( ( ( ( ph /\ x e. P ) /\ z e. P ) /\ ( A = ( x L z ) /\ x =/= z ) ) /\ X =/= x ) -> G e. TarskiG )
18		simp-4r 807	. . . . . 6 |- ( ( ( ( ( ph /\ x e. P ) /\ z e. P ) /\ ( A = ( x L z ) /\ x =/= z ) ) /\ X =/= x ) -> x e. P )
19		simpllr 799	. . . . . 6 |- ( ( ( ( ( ph /\ x e. P ) /\ z e. P ) /\ ( A = ( x L z ) /\ x =/= z ) ) /\ X =/= x ) -> z e. P )
20		simplrr 801	. . . . . 6 |- ( ( ( ( ( ph /\ x e. P ) /\ z e. P ) /\ ( A = ( x L z ) /\ x =/= z ) ) /\ X =/= x ) -> x =/= z )
21	1, 2, 3, 17, 18, 19, 20	tglinerflx1 25528	. . . . 5 |- ( ( ( ( ( ph /\ x e. P ) /\ z e. P ) /\ ( A = ( x L z ) /\ x =/= z ) ) /\ X =/= x ) -> x e. ( x L z ) )
22		simplrl 800	. . . . 5 |- ( ( ( ( ( ph /\ x e. P ) /\ z e. P ) /\ ( A = ( x L z ) /\ x =/= z ) ) /\ X =/= x ) -> A = ( x L z ) )
23	21, 22	eleqtrrd 2704	. . . 4 |- ( ( ( ( ( ph /\ x e. P ) /\ z e. P ) /\ ( A = ( x L z ) /\ x =/= z ) ) /\ X =/= x ) -> x e. A )
24		simpr 477	. . . 4 |- ( ( ( ( ( ph /\ x e. P ) /\ z e. P ) /\ ( A = ( x L z ) /\ x =/= z ) ) /\ X =/= x ) -> X =/= x )
25		neeq2 2857	. . . . 5 |- ( y = x -> ( X =/= y <-> X =/= x ) )
26	25	rspcev 3309	. . . 4 |- ( ( x e. A /\ X =/= x ) -> E. y e. A X =/= y )
27	23, 24, 26	syl2anc 693	. . 3 |- ( ( ( ( ( ph /\ x e. P ) /\ z e. P ) /\ ( A = ( x L z ) /\ x =/= z ) ) /\ X =/= x ) -> E. y e. A X =/= y )
28	16, 27	pm2.61dane 2881	. 2 |- ( ( ( ( ph /\ x e. P ) /\ z e. P ) /\ ( A = ( x L z ) /\ x =/= z ) ) -> E. y e. A X =/= y )
29		tglnpt2.a	. . 3 |- ( ph -> A e. ran L )
30	1, 2, 3, 4, 29	tgisline 25522	. 2 |- ( ph -> E. x e. P E. z e. P ( A = ( x L z ) /\ x =/= z ) )
31	28, 30	r19.29vva 3081	1 |- ( ph -> E. y e. A X =/= y )

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-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:  perpneq  25609  perpdrag  25620  oppperpex  25645  lnperpex  25695