Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > f1osng | Unicode version |
Description: A singleton of an ordered pair is one-to-one onto function. (Contributed by Mario Carneiro, 12-Jan-2013.) |
Ref | Expression |
---|---|
f1osng |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sneq 3386 | . . . 4 | |
2 | f1oeq2 5118 | . . . 4 | |
3 | 1, 2 | syl 14 | . . 3 |
4 | opeq1 3549 | . . . . 5 | |
5 | 4 | sneqd 3388 | . . . 4 |
6 | f1oeq1 5117 | . . . 4 | |
7 | 5, 6 | syl 14 | . . 3 |
8 | 3, 7 | bitrd 177 | . 2 |
9 | sneq 3386 | . . . 4 | |
10 | f1oeq3 5119 | . . . 4 | |
11 | 9, 10 | syl 14 | . . 3 |
12 | opeq2 3550 | . . . . 5 | |
13 | 12 | sneqd 3388 | . . . 4 |
14 | f1oeq1 5117 | . . . 4 | |
15 | 13, 14 | syl 14 | . . 3 |
16 | 11, 15 | bitrd 177 | . 2 |
17 | vex 2560 | . . 3 | |
18 | vex 2560 | . . 3 | |
19 | 17, 18 | f1osn 5166 | . 2 |
20 | 8, 16, 19 | vtocl2g 2617 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wceq 1243 wcel 1393 csn 3375 cop 3378 wf1o 4901 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 |
This theorem is referenced by: f1oprg 5168 fsnunf 5362 dif1en 6337 1fv 8996 |
Copyright terms: Public domain | W3C validator |