Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > nnpredcl | Unicode version |
Description: The predecessor of a natural number is a natural number. This theorem is most interesting when the natural number is a successor (as seen in theorems like onsucuni2 4523) but also holds when it is by uni0 3799. (Contributed by Jim Kingdon, 31-Jul-2022.) |
Ref | Expression |
---|---|
nnpredcl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unieq 3781 | . . . 4 | |
2 | uni0 3799 | . . . . 5 | |
3 | peano1 4553 | . . . . 5 | |
4 | 2, 3 | eqeltri 2230 | . . . 4 |
5 | 1, 4 | eqeltrdi 2248 | . . 3 |
6 | 5 | adantl 275 | . 2 |
7 | nnon 4569 | . . . . . 6 | |
8 | 7 | adantr 274 | . . . . 5 |
9 | simpr 109 | . . . . 5 | |
10 | onsucuni2 4523 | . . . . . . 7 | |
11 | 10 | ex 114 | . . . . . 6 |
12 | 11 | rexlimdvw 2578 | . . . . 5 |
13 | 8, 9, 12 | sylc 62 | . . . 4 |
14 | simpl 108 | . . . 4 | |
15 | 13, 14 | eqeltrd 2234 | . . 3 |
16 | peano2b 4574 | . . 3 | |
17 | 15, 16 | sylibr 133 | . 2 |
18 | nn0suc 4563 | . 2 | |
19 | 6, 17, 18 | mpjaodan 788 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1335 wcel 2128 wrex 2436 c0 3394 cuni 3772 con0 4323 csuc 4325 com 4549 |
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 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-iinf 4547 |
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-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 3773 df-int 3808 df-tr 4063 df-iord 4326 df-on 4328 df-suc 4331 df-iom 4550 |
This theorem is referenced by: nnpredlt 4583 omp1eomlem 7038 ctmlemr 7052 nnnninfeq2 7072 nninfisollemne 7074 nninfisol 7076 nnsf 13577 peano4nninf 13578 |
Copyright terms: Public domain | W3C validator |