ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nnpredcl Unicode version

Theorem nnpredcl 4531
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 4474) but also holds when it is  (/) by uni0 3758. (Contributed by Jim Kingdon, 31-Jul-2022.)
Assertion
Ref Expression
nnpredcl  |-  ( A  e.  om  ->  U. A  e.  om )

Proof of Theorem nnpredcl
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 unieq 3740 . . . 4  |-  ( A  =  (/)  ->  U. A  =  U. (/) )
2 uni0 3758 . . . . 5  |-  U. (/)  =  (/)
3 peano1 4503 . . . . 5  |-  (/)  e.  om
42, 3eqeltri 2210 . . . 4  |-  U. (/)  e.  om
51, 4syl6eqel 2228 . . 3  |-  ( A  =  (/)  ->  U. A  e.  om )
65adantl 275 . 2  |-  ( ( A  e.  om  /\  A  =  (/) )  ->  U. A  e.  om )
7 nnon 4518 . . . . . 6  |-  ( A  e.  om  ->  A  e.  On )
87adantr 274 . . . . 5  |-  ( ( A  e.  om  /\  E. x  e.  om  A  =  suc  x )  ->  A  e.  On )
9 simpr 109 . . . . 5  |-  ( ( A  e.  om  /\  E. x  e.  om  A  =  suc  x )  ->  E. x  e.  om  A  =  suc  x )
10 onsucuni2 4474 . . . . . . 7  |-  ( ( A  e.  On  /\  A  =  suc  x )  ->  suc  U. A  =  A )
1110ex 114 . . . . . 6  |-  ( A  e.  On  ->  ( A  =  suc  x  ->  suc  U. A  =  A ) )
1211rexlimdvw 2551 . . . . 5  |-  ( A  e.  On  ->  ( E. x  e.  om  A  =  suc  x  ->  suc  U. A  =  A ) )
138, 9, 12sylc 62 . . . 4  |-  ( ( A  e.  om  /\  E. x  e.  om  A  =  suc  x )  ->  suc  U. A  =  A )
14 simpl 108 . . . 4  |-  ( ( A  e.  om  /\  E. x  e.  om  A  =  suc  x )  ->  A  e.  om )
1513, 14eqeltrd 2214 . . 3  |-  ( ( A  e.  om  /\  E. x  e.  om  A  =  suc  x )  ->  suc  U. A  e.  om )
16 peano2b 4523 . . 3  |-  ( U. A  e.  om  <->  suc  U. A  e.  om )
1715, 16sylibr 133 . 2  |-  ( ( A  e.  om  /\  E. x  e.  om  A  =  suc  x )  ->  U. A  e.  om )
18 nn0suc 4513 . 2  |-  ( A  e.  om  ->  ( A  =  (/)  \/  E. x  e.  om  A  =  suc  x ) )
196, 17, 18mpjaodan 787 1  |-  ( A  e.  om  ->  U. A  e.  om )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480   E.wrex 2415   (/)c0 3358   U.cuni 3731   Oncon0 4280   suc csuc 4282   omcom 4499
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 2119  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-iinf 4497
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-uni 3732  df-int 3767  df-tr 4022  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500
This theorem is referenced by:  omp1eomlem  6972  ctmlemr  6986  nnsf  13188  peano4nninf  13189
  Copyright terms: Public domain W3C validator