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Theorem nnpredcl 4582
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.)
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 3781 . . . 4  |-  ( A  =  (/)  ->  U. A  =  U. (/) )
2 uni0 3799 . . . . 5  |-  U. (/)  =  (/)
3 peano1 4553 . . . . 5  |-  (/)  e.  om
42, 3eqeltri 2230 . . . 4  |-  U. (/)  e.  om
51, 4eqeltrdi 2248 . . 3  |-  ( A  =  (/)  ->  U. A  e.  om )
65adantl 275 . 2  |-  ( ( A  e.  om  /\  A  =  (/) )  ->  U. A  e.  om )
7 nnon 4569 . . . . . 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 4523 . . . . . . 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 2578 . . . . 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 2234 . . 3  |-  ( ( A  e.  om  /\  E. x  e.  om  A  =  suc  x )  ->  suc  U. A  e.  om )
16 peano2b 4574 . . 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 4563 . 2  |-  ( A  e.  om  ->  ( A  =  (/)  \/  E. x  e.  om  A  =  suc  x ) )
196, 17, 18mpjaodan 788 1  |-  ( A  e.  om  ->  U. A  e.  om )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1335    e. wcel 2128   E.wrex 2436   (/)c0 3394   U.cuni 3772   Oncon0 4323   suc csuc 4325   omcom 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
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