Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > uzaddcl | Unicode version |
Description: Addition closure law for an upper set of integers. (Contributed by NM, 4-Jun-2006.) |
Ref | Expression |
---|---|
uzaddcl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 5540 | . . . . 5 | |
2 | 1 | eleq1d 2147 | . . . 4 |
3 | 2 | imbi2d 228 | . . 3 |
4 | oveq2 5540 | . . . . 5 | |
5 | 4 | eleq1d 2147 | . . . 4 |
6 | 5 | imbi2d 228 | . . 3 |
7 | oveq2 5540 | . . . . 5 | |
8 | 7 | eleq1d 2147 | . . . 4 |
9 | 8 | imbi2d 228 | . . 3 |
10 | oveq2 5540 | . . . . 5 | |
11 | 10 | eleq1d 2147 | . . . 4 |
12 | 11 | imbi2d 228 | . . 3 |
13 | eluzelcn 8630 | . . . . . 6 | |
14 | 13 | addid1d 7257 | . . . . 5 |
15 | 14 | eleq1d 2147 | . . . 4 |
16 | 15 | ibir 175 | . . 3 |
17 | nn0cn 8298 | . . . . . . . 8 | |
18 | ax-1cn 7069 | . . . . . . . . 9 | |
19 | addass 7103 | . . . . . . . . 9 | |
20 | 18, 19 | mp3an3 1257 | . . . . . . . 8 |
21 | 13, 17, 20 | syl2anr 284 | . . . . . . 7 |
22 | 21 | adantr 270 | . . . . . 6 |
23 | peano2uz 8671 | . . . . . . 7 | |
24 | 23 | adantl 271 | . . . . . 6 |
25 | 22, 24 | eqeltrrd 2156 | . . . . 5 |
26 | 25 | exp31 356 | . . . 4 |
27 | 26 | a2d 26 | . . 3 |
28 | 3, 6, 9, 12, 16, 27 | nn0ind 8461 | . 2 |
29 | 28 | impcom 123 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wceq 1284 wcel 1433 cfv 4922 (class class class)co 5532 cc 6979 cc0 6981 c1 6982 caddc 6984 cn0 8288 cuz 8619 |
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-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-addcom 7076 ax-addass 7078 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-0id 7084 ax-rnegex 7085 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-ltadd 7092 |
This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-inn 8040 df-n0 8289 df-z 8352 df-uz 8620 |
This theorem is referenced by: elfz0add 9134 zpnn0elfzo 9216 |
Copyright terms: Public domain | W3C validator |