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Mirrors > Home > ILE Home > Th. List > xnn0lenn0nn0 | Unicode version |
Description: An extended nonnegative integer which is less than or equal to a nonnegative integer is a nonnegative integer. (Contributed by AV, 24-Nov-2021.) |
Ref | Expression |
---|---|
xnn0lenn0nn0 | NN0* |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxnn0 9042 | . . 3 NN0* | |
2 | 2a1 25 | . . . 4 | |
3 | breq1 3932 | . . . . . . 7 | |
4 | 3 | adantr 274 | . . . . . 6 |
5 | nn0re 8986 | . . . . . . . . . 10 | |
6 | 5 | rexrd 7815 | . . . . . . . . 9 |
7 | xgepnf 9599 | . . . . . . . . 9 | |
8 | 6, 7 | syl 14 | . . . . . . . 8 |
9 | pnfnre 7807 | . . . . . . . . 9 | |
10 | eleq1 2202 | . . . . . . . . . . 11 | |
11 | nn0re 8986 | . . . . . . . . . . . 12 | |
12 | elnelall 2415 | . . . . . . . . . . . 12 | |
13 | 11, 12 | syl 14 | . . . . . . . . . . 11 |
14 | 10, 13 | syl6bi 162 | . . . . . . . . . 10 |
15 | 14 | com13 80 | . . . . . . . . 9 |
16 | 9, 15 | ax-mp 5 | . . . . . . . 8 |
17 | 8, 16 | sylbid 149 | . . . . . . 7 |
18 | 17 | adantl 275 | . . . . . 6 |
19 | 4, 18 | sylbid 149 | . . . . 5 |
20 | 19 | ex 114 | . . . 4 |
21 | 2, 20 | jaoi 705 | . . 3 |
22 | 1, 21 | sylbi 120 | . 2 NN0* |
23 | 22 | 3imp 1175 | 1 NN0* |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 697 w3a 962 wceq 1331 wcel 1480 wnel 2403 class class class wbr 3929 cr 7619 cpnf 7797 cxr 7799 cle 7801 cn0 8977 NN0*cxnn0 9040 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1re 7714 ax-addrcl 7717 ax-rnegex 7729 ax-pre-ltirr 7732 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-xp 4545 df-cnv 4547 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-inn 8721 df-n0 8978 df-xnn0 9041 |
This theorem is referenced by: xnn0le2is012 9649 |
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