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Theorem bj-nn0suc 10089
Description: Proof of (biconditional form of) nn0suc 4327 from the core axioms of CZF. See also bj-nn0sucALT 10103. As a characterization of the elements of  om, this could be labeled "elom". (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-nn0suc  |-  ( A  e.  om  <->  ( A  =  (/)  \/  E. x  e.  om  A  =  suc  x ) )
Distinct variable group:    x, A

Proof of Theorem bj-nn0suc
StepHypRef Expression
1 bj-nn0suc0 10075 . . 3  |-  ( A  e.  om  ->  ( A  =  (/)  \/  E. x  e.  A  A  =  suc  x ) )
2 bj-omtrans 10081 . . . . 5  |-  ( A  e.  om  ->  A  C_ 
om )
3 ssrexv 3005 . . . . 5  |-  ( A 
C_  om  ->  ( E. x  e.  A  A  =  suc  x  ->  E. x  e.  om  A  =  suc  x ) )
42, 3syl 14 . . . 4  |-  ( A  e.  om  ->  ( E. x  e.  A  A  =  suc  x  ->  E. x  e.  om  A  =  suc  x ) )
54orim2d 702 . . 3  |-  ( A  e.  om  ->  (
( A  =  (/)  \/ 
E. x  e.  A  A  =  suc  x )  ->  ( A  =  (/)  \/  E. x  e. 
om  A  =  suc  x ) ) )
61, 5mpd 13 . 2  |-  ( A  e.  om  ->  ( A  =  (/)  \/  E. x  e.  om  A  =  suc  x ) )
7 peano1 4317 . . . 4  |-  (/)  e.  om
8 eleq1 2100 . . . 4  |-  ( A  =  (/)  ->  ( A  e.  om  <->  (/)  e.  om ) )
97, 8mpbiri 157 . . 3  |-  ( A  =  (/)  ->  A  e. 
om )
10 bj-peano2 10063 . . . . 5  |-  ( x  e.  om  ->  suc  x  e.  om )
11 eleq1a 2109 . . . . . 6  |-  ( suc  x  e.  om  ->  ( A  =  suc  x  ->  A  e.  om )
)
1211imp 115 . . . . 5  |-  ( ( suc  x  e.  om  /\  A  =  suc  x
)  ->  A  e.  om )
1310, 12sylan 267 . . . 4  |-  ( ( x  e.  om  /\  A  =  suc  x )  ->  A  e.  om )
1413rexlimiva 2428 . . 3  |-  ( E. x  e.  om  A  =  suc  x  ->  A  e.  om )
159, 14jaoi 636 . 2  |-  ( ( A  =  (/)  \/  E. x  e.  om  A  =  suc  x )  ->  A  e.  om )
166, 15impbii 117 1  |-  ( A  e.  om  <->  ( A  =  (/)  \/  E. x  e.  om  A  =  suc  x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 98    \/ wo 629    = wceq 1243    e. wcel 1393   E.wrex 2307    C_ wss 2917   (/)c0 3224   suc csuc 4102   omcom 4313
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-nul 3883  ax-pr 3944  ax-un 4170  ax-bd0 9933  ax-bdim 9934  ax-bdan 9935  ax-bdor 9936  ax-bdn 9937  ax-bdal 9938  ax-bdex 9939  ax-bdeq 9940  ax-bdel 9941  ax-bdsb 9942  ax-bdsep 10004  ax-infvn 10066
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-sn 3381  df-pr 3382  df-uni 3581  df-int 3616  df-suc 4108  df-iom 4314  df-bdc 9961  df-bj-ind 10051
This theorem is referenced by:  bj-findis  10104
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