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Theorem bj-nn0suc 11859
Description: Proof of (biconditional form of) nn0suc 4419 from the core axioms of CZF. See also bj-nn0sucALT 11873. 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 11845 . . 3  |-  ( A  e.  om  ->  ( A  =  (/)  \/  E. x  e.  A  A  =  suc  x ) )
2 bj-omtrans 11851 . . . . 5  |-  ( A  e.  om  ->  A  C_ 
om )
3 ssrexv 3086 . . . . 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 737 . . 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 4409 . . . 4  |-  (/)  e.  om
8 eleq1 2150 . . . 4  |-  ( A  =  (/)  ->  ( A  e.  om  <->  (/)  e.  om ) )
97, 8mpbiri 166 . . 3  |-  ( A  =  (/)  ->  A  e. 
om )
10 bj-peano2 11834 . . . . 5  |-  ( x  e.  om  ->  suc  x  e.  om )
11 eleq1a 2159 . . . . . 6  |-  ( suc  x  e.  om  ->  ( A  =  suc  x  ->  A  e.  om )
)
1211imp 122 . . . . 5  |-  ( ( suc  x  e.  om  /\  A  =  suc  x
)  ->  A  e.  om )
1310, 12sylan 277 . . . 4  |-  ( ( x  e.  om  /\  A  =  suc  x )  ->  A  e.  om )
1413rexlimiva 2484 . . 3  |-  ( E. x  e.  om  A  =  suc  x  ->  A  e.  om )
159, 14jaoi 671 . 2  |-  ( ( A  =  (/)  \/  E. x  e.  om  A  =  suc  x )  ->  A  e.  om )
166, 15impbii 124 1  |-  ( A  e.  om  <->  ( A  =  (/)  \/  E. x  e.  om  A  =  suc  x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    \/ wo 664    = wceq 1289    e. wcel 1438   E.wrex 2360    C_ wss 2999   (/)c0 3286   suc csuc 4192   omcom 4405
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-nul 3965  ax-pr 4036  ax-un 4260  ax-bd0 11704  ax-bdim 11705  ax-bdan 11706  ax-bdor 11707  ax-bdn 11708  ax-bdal 11709  ax-bdex 11710  ax-bdeq 11711  ax-bdel 11712  ax-bdsb 11713  ax-bdsep 11775  ax-infvn 11836
This theorem depends on definitions:  df-bi 115  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-rab 2368  df-v 2621  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-sn 3452  df-pr 3453  df-uni 3654  df-int 3689  df-suc 4198  df-iom 4406  df-bdc 11732  df-bj-ind 11822
This theorem is referenced by:  bj-findis  11874
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