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Mirrors > Home > MPE Home > Th. List > oppne1 | Structured version Visualization version Unicode version |
Description: Points lying on opposite sides of a line cannot be on the line. (Contributed by Thierry Arnoux, 3-Mar-2020.) |
Ref | Expression |
---|---|
hpg.p | |
hpg.d | |
hpg.i | Itv |
hpg.o | |
opphl.l | LineG |
opphl.d | |
opphl.g | TarskiG |
oppcom.a | |
oppcom.b | |
oppcom.o |
Ref | Expression |
---|---|
oppne1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oppcom.o | . . 3 | |
2 | hpg.p | . . . 4 | |
3 | hpg.d | . . . 4 | |
4 | hpg.i | . . . 4 Itv | |
5 | hpg.o | . . . 4 | |
6 | oppcom.a | . . . 4 | |
7 | oppcom.b | . . . 4 | |
8 | 2, 3, 4, 5, 6, 7 | islnopp 25631 | . . 3 |
9 | 1, 8 | mpbid 222 | . 2 |
10 | 9 | simplld 791 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wa 384 wceq 1483 wcel 1990 wrex 2913 cdif 3571 class class class wbr 4653 copab 4712 crn 5115 cfv 5888 (class class class)co 6650 cbs 15857 cds 15950 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-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-rex 2918 df-rab 2921 df-v 3202 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-iota 5851 df-fv 5896 df-ov 6653 |
This theorem is referenced by: oppne3 25635 opptgdim2 25637 opphllem1 25639 opphllem2 25640 opphl 25646 hpgne1 25653 hpgne2 25654 lnopp2hpgb 25655 colopp 25661 |
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