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Theorem bj-nn0suc 15401
Description: Proof of (biconditional form of) nn0suc 4632 from the core axioms of CZF. See also bj-nn0sucALT 15415. 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 15387 . . 3  |-  ( A  e.  om  ->  ( A  =  (/)  \/  E. x  e.  A  A  =  suc  x ) )
2 bj-omtrans 15393 . . . . 5  |-  ( A  e.  om  ->  A  C_ 
om )
3 ssrexv 3244 . . . . 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 789 . . 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 4622 . . . 4  |-  (/)  e.  om
8 eleq1 2256 . . . 4  |-  ( A  =  (/)  ->  ( A  e.  om  <->  (/)  e.  om ) )
97, 8mpbiri 168 . . 3  |-  ( A  =  (/)  ->  A  e. 
om )
10 bj-peano2 15376 . . . . 5  |-  ( x  e.  om  ->  suc  x  e.  om )
11 eleq1a 2265 . . . . . 6  |-  ( suc  x  e.  om  ->  ( A  =  suc  x  ->  A  e.  om )
)
1211imp 124 . . . . 5  |-  ( ( suc  x  e.  om  /\  A  =  suc  x
)  ->  A  e.  om )
1310, 12sylan 283 . . . 4  |-  ( ( x  e.  om  /\  A  =  suc  x )  ->  A  e.  om )
1413rexlimiva 2606 . . 3  |-  ( E. x  e.  om  A  =  suc  x  ->  A  e.  om )
159, 14jaoi 717 . 2  |-  ( ( A  =  (/)  \/  E. x  e.  om  A  =  suc  x )  ->  A  e.  om )
166, 15impbii 126 1  |-  ( A  e.  om  <->  ( A  =  (/)  \/  E. x  e.  om  A  =  suc  x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    \/ wo 709    = wceq 1364    e. wcel 2164   E.wrex 2473    C_ wss 3153   (/)c0 3446   suc csuc 4394   omcom 4618
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-nul 4155  ax-pr 4238  ax-un 4462  ax-bd0 15250  ax-bdim 15251  ax-bdan 15252  ax-bdor 15253  ax-bdn 15254  ax-bdal 15255  ax-bdex 15256  ax-bdeq 15257  ax-bdel 15258  ax-bdsb 15259  ax-bdsep 15321  ax-infvn 15378
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-sn 3624  df-pr 3625  df-uni 3836  df-int 3871  df-suc 4400  df-iom 4619  df-bdc 15278  df-bj-ind 15364
This theorem is referenced by:  bj-findis  15416
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