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Theorem bj-nn0suc 13089
Description: Proof of (biconditional form of) nn0suc 4488 from the core axioms of CZF. See also bj-nn0sucALT 13103. 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 13075 . . 3  |-  ( A  e.  om  ->  ( A  =  (/)  \/  E. x  e.  A  A  =  suc  x ) )
2 bj-omtrans 13081 . . . . 5  |-  ( A  e.  om  ->  A  C_ 
om )
3 ssrexv 3132 . . . . 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 762 . . 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 4478 . . . 4  |-  (/)  e.  om
8 eleq1 2180 . . . 4  |-  ( A  =  (/)  ->  ( A  e.  om  <->  (/)  e.  om ) )
97, 8mpbiri 167 . . 3  |-  ( A  =  (/)  ->  A  e. 
om )
10 bj-peano2 13064 . . . . 5  |-  ( x  e.  om  ->  suc  x  e.  om )
11 eleq1a 2189 . . . . . 6  |-  ( suc  x  e.  om  ->  ( A  =  suc  x  ->  A  e.  om )
)
1211imp 123 . . . . 5  |-  ( ( suc  x  e.  om  /\  A  =  suc  x
)  ->  A  e.  om )
1310, 12sylan 281 . . . 4  |-  ( ( x  e.  om  /\  A  =  suc  x )  ->  A  e.  om )
1413rexlimiva 2521 . . 3  |-  ( E. x  e.  om  A  =  suc  x  ->  A  e.  om )
159, 14jaoi 690 . 2  |-  ( ( A  =  (/)  \/  E. x  e.  om  A  =  suc  x )  ->  A  e.  om )
166, 15impbii 125 1  |-  ( A  e.  om  <->  ( A  =  (/)  \/  E. x  e.  om  A  =  suc  x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    \/ wo 682    = wceq 1316    e. wcel 1465   E.wrex 2394    C_ wss 3041   (/)c0 3333   suc csuc 4257   omcom 4474
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-nul 4024  ax-pr 4101  ax-un 4325  ax-bd0 12938  ax-bdim 12939  ax-bdan 12940  ax-bdor 12941  ax-bdn 12942  ax-bdal 12943  ax-bdex 12944  ax-bdeq 12945  ax-bdel 12946  ax-bdsb 12947  ax-bdsep 13009  ax-infvn 13066
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-sn 3503  df-pr 3504  df-uni 3707  df-int 3742  df-suc 4263  df-iom 4475  df-bdc 12966  df-bj-ind 13052
This theorem is referenced by:  bj-findis  13104
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