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Theorem nnpredcl 4607
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.)
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 3805 . . . 4  |-  ( A  =  (/)  ->  U. A  =  U. (/) )
2 uni0 3823 . . . . 5  |-  U. (/)  =  (/)
3 peano1 4578 . . . . 5  |-  (/)  e.  om
42, 3eqeltri 2243 . . . 4  |-  U. (/)  e.  om
51, 4eqeltrdi 2261 . . 3  |-  ( A  =  (/)  ->  U. A  e.  om )
65adantl 275 . 2  |-  ( ( A  e.  om  /\  A  =  (/) )  ->  U. A  e.  om )
7 nnon 4594 . . . . . 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 4548 . . . . . . 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 2591 . . . . 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 2247 . . 3  |-  ( ( A  e.  om  /\  E. x  e.  om  A  =  suc  x )  ->  suc  U. A  e.  om )
16 peano2b 4599 . . 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 4588 . 2  |-  ( A  e.  om  ->  ( A  =  (/)  \/  E. x  e.  om  A  =  suc  x ) )
196, 17, 18mpjaodan 793 1  |-  ( A  e.  om  ->  U. A  e.  om )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1348    e. wcel 2141   E.wrex 2449   (/)c0 3414   U.cuni 3796   Oncon0 4348   suc csuc 4350   omcom 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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