Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bj-nn0suc Unicode version

Theorem bj-nn0suc 14598
Description: Proof of (biconditional form of) nn0suc 4603 from the core axioms of CZF. See also bj-nn0sucALT 14612. 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 14584 . . 3  |-  ( A  e.  om  ->  ( A  =  (/)  \/  E. x  e.  A  A  =  suc  x ) )
2 bj-omtrans 14590 . . . . 5  |-  ( A  e.  om  ->  A  C_ 
om )
3 ssrexv 3220 . . . . 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 788 . . 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 4593 . . . 4  |-  (/)  e.  om
8 eleq1 2240 . . . 4  |-  ( A  =  (/)  ->  ( A  e.  om  <->  (/)  e.  om ) )
97, 8mpbiri 168 . . 3  |-  ( A  =  (/)  ->  A  e. 
om )
10 bj-peano2 14573 . . . . 5  |-  ( x  e.  om  ->  suc  x  e.  om )
11 eleq1a 2249 . . . . . 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 2589 . . 3  |-  ( E. x  e.  om  A  =  suc  x  ->  A  e.  om )
159, 14jaoi 716 . 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 708    = wceq 1353    e. wcel 2148   E.wrex 2456    C_ wss 3129   (/)c0 3422   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-nul 4129  ax-pr 4209  ax-un 4433  ax-bd0 14447  ax-bdim 14448  ax-bdan 14449  ax-bdor 14450  ax-bdn 14451  ax-bdal 14452  ax-bdex 14453  ax-bdeq 14454  ax-bdel 14455  ax-bdsb 14456  ax-bdsep 14518  ax-infvn 14575
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-sn 3598  df-pr 3599  df-uni 3810  df-int 3845  df-suc 4371  df-iom 4590  df-bdc 14475  df-bj-ind 14561
This theorem is referenced by:  bj-findis  14613
  Copyright terms: Public domain W3C validator