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Mirrors > Home > ILE Home > Th. List > nnpredlt | Unicode version |
Description: The predecessor (see nnpredcl 4583) of a nonzero natural number is less than (see df-iord 4327) that number. (Contributed by Jim Kingdon, 14-Sep-2024.) |
Ref | Expression |
---|---|
nnpredlt |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnpredcl 4583 | . . . 4 | |
2 | 1 | adantr 274 | . . 3 |
3 | sucidg 4377 | . . 3 | |
4 | 2, 3 | syl 14 | . 2 |
5 | nnsucpred 4577 | . 2 | |
6 | 4, 5 | eleqtrd 2236 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wcel 2128 wne 2327 c0 3394 cuni 3773 csuc 4326 com 4550 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4083 ax-nul 4091 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-iinf 4548 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-v 2714 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-uni 3774 df-int 3809 df-tr 4064 df-iord 4327 df-on 4329 df-suc 4332 df-iom 4551 |
This theorem is referenced by: nninfisollemne 7075 |
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