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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 4548) but also holds when it is by uni0 3823. (Contributed by Jim Kingdon, 31-Jul-2022.) |
Ref | Expression |
---|---|
nnpredcl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unieq 3805 | . . . 4 | |
2 | uni0 3823 | . . . . 5 | |
3 | peano1 4578 | . . . . 5 | |
4 | 2, 3 | eqeltri 2243 | . . . 4 |
5 | 1, 4 | eqeltrdi 2261 | . . 3 |
6 | 5 | adantl 275 | . 2 |
7 | nnon 4594 | . . . . . 6 | |
8 | 7 | adantr 274 | . . . . 5 |
9 | simpr 109 | . . . . 5 | |
10 | onsucuni2 4548 | . . . . . . 7 | |
11 | 10 | ex 114 | . . . . . 6 |
12 | 11 | rexlimdvw 2591 | . . . . 5 |
13 | 8, 9, 12 | sylc 62 | . . . 4 |
14 | simpl 108 | . . . 4 | |
15 | 13, 14 | eqeltrd 2247 | . . 3 |
16 | peano2b 4599 | . . 3 | |
17 | 15, 16 | sylibr 133 | . 2 |
18 | nn0suc 4588 | . 2 | |
19 | 6, 17, 18 | mpjaodan 793 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1348 wcel 2141 wrex 2449 c0 3414 cuni 3796 con0 4348 csuc 4350 com 4574 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-iinf 4572 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-uni 3797 df-int 3832 df-tr 4088 df-iord 4351 df-on 4353 df-suc 4356 df-iom 4575 |
This theorem is referenced by: nnpredlt 4608 omp1eomlem 7071 ctmlemr 7085 nnnninfeq2 7105 nninfisollemne 7107 nninfisol 7109 nnsf 14038 peano4nninf 14039 |
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