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Mirrors > Home > ILE Home > Th. List > 0nelop | Unicode version |
Description: A property of ordered pairs. (Contributed by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
0nelop |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 19 | . . . 4 | |
2 | oprcl 3594 | . . . . 5 | |
3 | dfopg 3568 | . . . . 5 | |
4 | 2, 3 | syl 14 | . . . 4 |
5 | 1, 4 | eleqtrd 2157 | . . 3 |
6 | elpri 3421 | . . 3 | |
7 | 5, 6 | syl 14 | . 2 |
8 | 2 | simpld 110 | . . . . . 6 |
9 | snnzg 3507 | . . . . . 6 | |
10 | 8, 9 | syl 14 | . . . . 5 |
11 | 10 | necomd 2331 | . . . 4 |
12 | prnzg 3514 | . . . . . 6 | |
13 | 8, 12 | syl 14 | . . . . 5 |
14 | 13 | necomd 2331 | . . . 4 |
15 | 11, 14 | jca 300 | . . 3 |
16 | neanior 2332 | . . 3 | |
17 | 15, 16 | sylib 120 | . 2 |
18 | 7, 17 | pm2.65i 600 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wa 102 wo 661 wceq 1284 wcel 1433 wne 2245 cvv 2601 c0 3251 csn 3398 cpr 3399 cop 3401 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-v 2603 df-dif 2975 df-un 2977 df-nul 3252 df-sn 3404 df-pr 3405 df-op 3407 |
This theorem is referenced by: 0nelelxp 4391 |
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