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Theorem nnsucpred 4616
Description: The successor of the precedessor of a nonzero natural number. (Contributed by Jim Kingdon, 31-Jul-2022.)
Assertion
Ref Expression
nnsucpred  |-  ( ( A  e.  om  /\  A  =/=  (/) )  ->  suc  U. A  =  A )

Proof of Theorem nnsucpred
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 nnsuc 4615 . 2  |-  ( ( A  e.  om  /\  A  =/=  (/) )  ->  E. x  e.  om  A  =  suc  x )
2 nnon 4609 . . . 4  |-  ( A  e.  om  ->  A  e.  On )
32ad2antrr 488 . . 3  |-  ( ( ( A  e.  om  /\  A  =/=  (/) )  /\  ( x  e.  om  /\  A  =  suc  x
) )  ->  A  e.  On )
4 simprr 531 . . 3  |-  ( ( ( A  e.  om  /\  A  =/=  (/) )  /\  ( x  e.  om  /\  A  =  suc  x
) )  ->  A  =  suc  x )
5 onsucuni2 4563 . . 3  |-  ( ( A  e.  On  /\  A  =  suc  x )  ->  suc  U. A  =  A )
63, 4, 5syl2anc 411 . 2  |-  ( ( ( A  e.  om  /\  A  =/=  (/) )  /\  ( x  e.  om  /\  A  =  suc  x
) )  ->  suc  U. A  =  A )
71, 6rexlimddv 2599 1  |-  ( ( A  e.  om  /\  A  =/=  (/) )  ->  suc  U. A  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148    =/= wne 2347   (/)c0 3422   U.cuni 3809   Oncon0 4363   suc csuc 4365   omcom 4589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-iinf 4587
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-uni 3810  df-int 3845  df-tr 4102  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590
This theorem is referenced by:  nnpredlt  4623  omp1eomlem  7092  nnnninfeq2  7126  nnsf  14636
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