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Theorem nnpredcl 11536
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 4370) but also holds when it is  (/) by uni0 3675. (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 3657 . . . 4  |-  ( A  =  (/)  ->  U. A  =  U. (/) )
2 uni0 3675 . . . . 5  |-  U. (/)  =  (/)
3 peano1 4399 . . . . 5  |-  (/)  e.  om
42, 3eqeltri 2160 . . . 4  |-  U. (/)  e.  om
51, 4syl6eqel 2178 . . 3  |-  ( A  =  (/)  ->  U. A  e.  om )
65adantl 271 . 2  |-  ( ( A  e.  om  /\  A  =  (/) )  ->  U. A  e.  om )
7 nnon 4414 . . . . . 6  |-  ( A  e.  om  ->  A  e.  On )
87adantr 270 . . . . 5  |-  ( ( A  e.  om  /\  E. x  e.  om  A  =  suc  x )  ->  A  e.  On )
9 simpr 108 . . . . 5  |-  ( ( A  e.  om  /\  E. x  e.  om  A  =  suc  x )  ->  E. x  e.  om  A  =  suc  x )
10 onsucuni2 4370 . . . . . . 7  |-  ( ( A  e.  On  /\  A  =  suc  x )  ->  suc  U. A  =  A )
1110ex 113 . . . . . 6  |-  ( A  e.  On  ->  ( A  =  suc  x  ->  suc  U. A  =  A ) )
1211rexlimdvw 2492 . . . . 5  |-  ( A  e.  On  ->  ( E. x  e.  om  A  =  suc  x  ->  suc  U. A  =  A ) )
138, 9, 12sylc 61 . . . 4  |-  ( ( A  e.  om  /\  E. x  e.  om  A  =  suc  x )  ->  suc  U. A  =  A )
14 simpl 107 . . . 4  |-  ( ( A  e.  om  /\  E. x  e.  om  A  =  suc  x )  ->  A  e.  om )
1513, 14eqeltrd 2164 . . 3  |-  ( ( A  e.  om  /\  E. x  e.  om  A  =  suc  x )  ->  suc  U. A  e.  om )
16 peano2b 4419 . . 3  |-  ( U. A  e.  om  <->  suc  U. A  e.  om )
1715, 16sylibr 132 . 2  |-  ( ( A  e.  om  /\  E. x  e.  om  A  =  suc  x )  ->  U. A  e.  om )
18 nn0suc 4409 . 2  |-  ( A  e.  om  ->  ( A  =  (/)  \/  E. x  e.  om  A  =  suc  x ) )
196, 17, 18mpjaodan 747 1  |-  ( A  e.  om  ->  U. A  e.  om )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438   E.wrex 2360   (/)c0 3284   U.cuni 3648   Oncon0 4181   suc csuc 4183   omcom 4395
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-iinf 4393
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-uni 3649  df-int 3684  df-tr 3929  df-iord 4184  df-on 4186  df-suc 4189  df-iom 4396
This theorem is referenced by:  nnsf  11541  peano4nninf  11542
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