Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > cvrnbtwn2 | Structured version Visualization version Unicode version |
Description: The covers relation implies no in-betweenness. (cvnbtwn2 29146 analog.) (Contributed by NM, 17-Nov-2011.) |
Ref | Expression |
---|---|
cvrletr.b | |
cvrletr.l | |
cvrletr.s | |
cvrletr.c |
Ref | Expression |
---|---|
cvrnbtwn2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cvrletr.b | . . . . . 6 | |
2 | cvrletr.s | . . . . . 6 | |
3 | cvrletr.c | . . . . . 6 | |
4 | 1, 2, 3 | cvrnbtwn 34558 | . . . . 5 |
5 | 4 | 3expia 1267 | . . . 4 |
6 | iman 440 | . . . . 5 | |
7 | simpl 473 | . . . . . . . . . 10 | |
8 | simpr3 1069 | . . . . . . . . . 10 | |
9 | simpr2 1068 | . . . . . . . . . 10 | |
10 | cvrletr.l | . . . . . . . . . . 11 | |
11 | 10, 2 | pltval 16960 | . . . . . . . . . 10 |
12 | 7, 8, 9, 11 | syl3anc 1326 | . . . . . . . . 9 |
13 | df-ne 2795 | . . . . . . . . . 10 | |
14 | 13 | anbi2i 730 | . . . . . . . . 9 |
15 | 12, 14 | syl6bb 276 | . . . . . . . 8 |
16 | 15 | anbi2d 740 | . . . . . . 7 |
17 | anass 681 | . . . . . . 7 | |
18 | 16, 17 | syl6rbbr 279 | . . . . . 6 |
19 | 18 | notbid 308 | . . . . 5 |
20 | 6, 19 | syl5rbb 273 | . . . 4 |
21 | 5, 20 | sylibd 229 | . . 3 |
22 | 21 | 3impia 1261 | . 2 |
23 | 1, 2, 3 | cvrlt 34557 | . . . . . . 7 |
24 | 23 | ex 450 | . . . . . 6 |
25 | 24 | 3adant3r3 1276 | . . . . 5 |
26 | 25 | 3impia 1261 | . . . 4 |
27 | breq2 4657 | . . . 4 | |
28 | 26, 27 | syl5ibrcom 237 | . . 3 |
29 | 1, 10 | posref 16951 | . . . . . 6 |
30 | 29 | 3ad2antr2 1227 | . . . . 5 |
31 | breq1 4656 | . . . . 5 | |
32 | 30, 31 | syl5ibrcom 237 | . . . 4 |
33 | 32 | 3adant3 1081 | . . 3 |
34 | 28, 33 | jcad 555 | . 2 |
35 | 22, 34 | impbid 202 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 wa 384 w3a 1037 wceq 1483 wcel 1990 wne 2794 class class class wbr 4653 cfv 5888 cbs 15857 cple 15948 cpo 16940 cplt 16941 ccvr 34549 |
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-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-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-preset 16928 df-poset 16946 df-plt 16958 df-covers 34553 |
This theorem is referenced by: cvrval3 34699 cvrexchlem 34705 |
Copyright terms: Public domain | W3C validator |