Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > nn0ennn | Unicode version |
Description: The nonnegative integers are equinumerous to the positive integers. (Contributed by NM, 19-Jul-2004.) |
Ref | Expression |
---|---|
nn0ennn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0ex 8187 | . 2 | |
2 | nnex 7920 | . 2 | |
3 | nn0p1nn 8221 | . 2 | |
4 | nnm1nn0 8223 | . 2 | |
5 | nncn 7922 | . . 3 | |
6 | nn0cn 8191 | . . 3 | |
7 | ax-1cn 6977 | . . . . . 6 | |
8 | subadd 7214 | . . . . . 6 | |
9 | 7, 8 | mp3an2 1220 | . . . . 5 |
10 | eqcom 2042 | . . . . 5 | |
11 | eqcom 2042 | . . . . 5 | |
12 | 9, 10, 11 | 3bitr4g 212 | . . . 4 |
13 | addcom 7150 | . . . . . . 7 | |
14 | 7, 13 | mpan 400 | . . . . . 6 |
15 | 14 | eqeq2d 2051 | . . . . 5 |
16 | 15 | adantl 262 | . . . 4 |
17 | 12, 16 | bitrd 177 | . . 3 |
18 | 5, 6, 17 | syl2anr 274 | . 2 |
19 | 1, 2, 3, 4, 18 | en3i 6251 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 97 wb 98 wceq 1243 wcel 1393 class class class wbr 3764 (class class class)co 5512 cen 6219 cc 6887 c1 6890 caddc 6892 cmin 7182 cn 7914 cn0 8181 |
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-in1 544 ax-in2 545 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-13 1404 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 ax-un 4170 ax-setind 4262 ax-cnex 6975 ax-resscn 6976 ax-1cn 6977 ax-1re 6978 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-addcom 6984 ax-addass 6986 ax-distr 6988 ax-i2m1 6989 ax-0id 6992 ax-rnegex 6993 ax-cnre 6995 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 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-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-riota 5468 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-en 6222 df-sub 7184 df-inn 7915 df-n0 8182 |
This theorem is referenced by: nnenom 9210 |
Copyright terms: Public domain | W3C validator |