tgisline - Metamath Proof Explorer

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Theorem tgisline 25522

Description: The property of being a proper line, generated by two distinct points. (Contributed by Thierry Arnoux, 25-May-2019.)

**Hypotheses**
Ref	Expression
tglineelsb2.p
tglineelsb2.i	Itv
tglineelsb2.l	LineG
tglineelsb2.g	TarskiG
tgisline.1

**Assertion**
Ref	Expression
tgisline	$/\$

Distinct variable groups:

Allowed substitution hints:

(

)

Proof of Theorem tgisline Dummy variable

is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tglineelsb2.p	. . . . . 6
2		tglineelsb2.l	. . . . . 6 LineG
3		tglineelsb2.i	. . . . . 6 Itv
4		tglineelsb2.g	. . . . . . 7 TarskiG
5	4	adantr 481	. . . . . 6 $/\$ $/\$ $\$ TarskiG
6		simprl 794	. . . . . 6 $/\$ $/\$ $\$
7		simprr 796	. . . . . . 7 $/\$ $/\$ $\$ $\$
8	7	eldifad 3586	. . . . . 6 $/\$ $/\$ $\$
9		eldifsn 4317	. . . . . . . . 9 $\$ $/\$
10	7, 9	sylib 208	. . . . . . . 8 $/\$ $/\$ $\$ $/\$
11	10	simprd 479	. . . . . . 7 $/\$ $/\$ $\$
12	11	necomd 2849	. . . . . 6 $/\$ $/\$ $\$
13	1, 2, 3, 5, 6, 8, 12	tglngval 25446	. . . . 5 $/\$ $/\$ $\$ $\/$ $\/$
14	13, 12	jca 554	. . . 4 $/\$ $/\$ $\$ $\/$ $\/$ $/\$
15	14	ralrimivva 2971	. . 3 $\$ $\/$ $\/$ $/\$
16		tgisline.1	. . . . 5
17	1, 2, 3	tglng 25441	. . . . . . 7 TarskiG $\$ $\/$ $\/$
18	4, 17	syl 17	. . . . . 6 $\$ $\/$ $\/$
19	18	rneqd 5353	. . . . 5 $\$ $\/$ $\/$
20	16, 19	eleqtrd 2703	. . . 4 $\$ $\/$ $\/$
21		eqid 2622	. . . . . 6 $\$ $\/$ $\/$ $\$ $\/$ $\/$
22	21	elrnmpt2g 6772	. . . . 5 $\$ $\/$ $\/$ $\$ $\/$ $\/$
23	16, 22	syl 17	. . . 4 $\$ $\/$ $\/$ $\$ $\/$ $\/$
24	20, 23	mpbid 222	. . 3 $\$ $\/$ $\/$
25	15, 24	r19.29d2r 3080	. 2 $\$ $\/$ $\/$ $/\$ $/\$ $\/$ $\/$
26		difss 3737	. . . 4 $\$
27		simpr 477	. . . . . . 7 $\/$ $\/$ $/\$ $/\$ $\/$ $\/$ $\/$ $\/$
28		simpll 790	. . . . . . 7 $\/$ $\/$ $/\$ $/\$ $\/$ $\/$ $\/$ $\/$
29	27, 28	eqtr4d 2659	. . . . . 6 $\/$ $\/$ $/\$ $/\$ $\/$ $\/$
30		simplr 792	. . . . . 6 $\/$ $\/$ $/\$ $/\$ $\/$ $\/$
31	29, 30	jca 554	. . . . 5 $\/$ $\/$ $/\$ $/\$ $\/$ $\/$ $/\$
32	31	reximi 3011	. . . 4 $\$ $\/$ $\/$ $/\$ $/\$ $\/$ $\/$ $\$ $/\$
33		ssrexv 3667	. . . 4 $\$ $\$ $/\$ $/\$
34	26, 32, 33	mpsyl 68	. . 3 $\$ $\/$ $\/$ $/\$ $/\$ $\/$ $\/$ $/\$
35	34	reximi 3011	. 2 $\$ $\/$ $\/$ $/\$ $/\$ $\/$ $\/$ $/\$
36	25, 35	syl 17	1 $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $/\$ wa 384 $\/$ w3o 1036

wceq 1483

wcel 1990

wne 2794

wrex 2913

crab 2916 $\$ cdif 3571

wss 3574

csn 4177

crn 5115

cfv 5888 (class class class)co 6650

cmpt2 6652

cbs 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-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-trkg 25352

This theorem is referenced by: tglnne 25523 tglndim0 25524 tglinethru 25531 tglnne0 25535 tglnpt2 25536 footex 25613 opptgdim2 25637

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