Description: If two different points X and Y in a real Euclidean space of dimension 2 are different, then at least one difference of two corresponding coordinates is not 0. (Contributed by AV, 26-Feb-2023)
Ref | Expression | ||
---|---|---|---|
Hypotheses | rrx2pnecoorneor.i | |
|
rrx2pnecoorneor.b | |
||
rrx2pnedifcoorneor.a | |
||
rrx2pnedifcoorneorr.b | |
||
Assertion | rrx2pnedifcoorneorr | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rrx2pnecoorneor.i | |
|
2 | rrx2pnecoorneor.b | |
|
3 | rrx2pnedifcoorneor.a | |
|
4 | rrx2pnedifcoorneorr.b | |
|
5 | eqid | |
|
6 | 1 2 3 5 | rrx2pnedifcoorneor | |
7 | eqcom | |
|
8 | 7 | a1i | |
9 | 1 2 | rrx2pyel | |
10 | 9 | recnd | |
11 | 1 2 | rrx2pyel | |
12 | 11 | recnd | |
13 | 10 12 | anim12i | |
14 | 13 | ancomd | |
15 | 14 | 3adant3 | |
16 | subeq0 | |
|
17 | 15 16 | syl | |
18 | 13 | 3adant3 | |
19 | subeq0 | |
|
20 | 18 19 | syl | |
21 | 8 17 20 | 3bitr4d | |
22 | 4 | eqcomi | |
23 | 22 | eqeq1i | |
24 | 21 23 | bitrdi | |
25 | 24 | necon3bid | |
26 | 25 | orbi2d | |
27 | 6 26 | mpbid | |