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Theorem bj-nn0suc0 12733
Description: Constructive proof of a variant of nn0suc 4456. For a constructive proof of nn0suc 4456, see bj-nn0suc 12747. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-nn0suc0  |-  ( A  e.  om  ->  ( A  =  (/)  \/  E. x  e.  A  A  =  suc  x ) )
Distinct variable group:    x, A

Proof of Theorem bj-nn0suc0
Dummy variables  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq1 2106 . . 3  |-  ( y  =  A  ->  (
y  =  (/)  <->  A  =  (/) ) )
2 eqeq1 2106 . . . 4  |-  ( y  =  A  ->  (
y  =  suc  x  <->  A  =  suc  x ) )
32rexeqbi1dv 2593 . . 3  |-  ( y  =  A  ->  ( E. x  e.  y 
y  =  suc  x  <->  E. x  e.  A  A  =  suc  x ) )
41, 3orbi12d 748 . 2  |-  ( y  =  A  ->  (
( y  =  (/)  \/ 
E. x  e.  y  y  =  suc  x
)  <->  ( A  =  (/)  \/  E. x  e.  A  A  =  suc  x ) ) )
5 tru 1303 . . 3  |- T.
6 a1tru 1315 . . . 4  |-  ( T. 
-> T.  )
76rgenw 2446 . . 3  |-  A. z  e.  om  ( T.  -> T.  )
8 bdeq0 12646 . . . . 5  |- BOUNDED  y  =  (/)
9 bdeqsuc 12660 . . . . . 6  |- BOUNDED  y  =  suc  x
109ax-bdex 12598 . . . . 5  |- BOUNDED  E. x  e.  y  y  =  suc  x
118, 10ax-bdor 12595 . . . 4  |- BOUNDED  ( y  =  (/)  \/ 
E. x  e.  y  y  =  suc  x
)
12 nfv 1476 . . . 4  |-  F/ y T.
13 orc 674 . . . . 5  |-  ( y  =  (/)  ->  ( y  =  (/)  \/  E. x  e.  y  y  =  suc  x ) )
1413a1d 22 . . . 4  |-  ( y  =  (/)  ->  ( T. 
->  ( y  =  (/)  \/ 
E. x  e.  y  y  =  suc  x
) ) )
15 a1tru 1315 . . . . 5  |-  ( -.  ( y  =  z  ->  -.  ( y  =  (/)  \/  E. x  e.  y  y  =  suc  x ) )  -> T.  )
1615expi 607 . . . 4  |-  ( y  =  z  ->  (
( y  =  (/)  \/ 
E. x  e.  y  y  =  suc  x
)  -> T.  )
)
17 vex 2644 . . . . . . . . 9  |-  z  e. 
_V
1817sucid 4277 . . . . . . . 8  |-  z  e. 
suc  z
19 eleq2 2163 . . . . . . . 8  |-  ( y  =  suc  z  -> 
( z  e.  y  <-> 
z  e.  suc  z
) )
2018, 19mpbiri 167 . . . . . . 7  |-  ( y  =  suc  z  -> 
z  e.  y )
21 suceq 4262 . . . . . . . . 9  |-  ( x  =  z  ->  suc  x  =  suc  z )
2221eqeq2d 2111 . . . . . . . 8  |-  ( x  =  z  ->  (
y  =  suc  x  <->  y  =  suc  z ) )
2322rspcev 2744 . . . . . . 7  |-  ( ( z  e.  y  /\  y  =  suc  z )  ->  E. x  e.  y  y  =  suc  x
)
2420, 23mpancom 416 . . . . . 6  |-  ( y  =  suc  z  ->  E. x  e.  y 
y  =  suc  x
)
2524olcd 694 . . . . 5  |-  ( y  =  suc  z  -> 
( y  =  (/)  \/ 
E. x  e.  y  y  =  suc  x
) )
2625a1d 22 . . . 4  |-  ( y  =  suc  z  -> 
( T.  ->  (
y  =  (/)  \/  E. x  e.  y  y  =  suc  x ) ) )
2711, 12, 12, 12, 14, 16, 26bj-bdfindis 12730 . . 3  |-  ( ( T.  /\  A. z  e.  om  ( T.  -> T.  ) )  ->  A. y  e.  om  ( y  =  (/)  \/  E. x  e.  y  y  =  suc  x ) )
285, 7, 27mp2an 420 . 2  |-  A. y  e.  om  ( y  =  (/)  \/  E. x  e.  y  y  =  suc  x )
294, 28vtoclri 2716 1  |-  ( A  e.  om  ->  ( A  =  (/)  \/  E. x  e.  A  A  =  suc  x ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 670    = wceq 1299   T. wtru 1300    e. wcel 1448   A.wral 2375   E.wrex 2376   (/)c0 3310   suc csuc 4225   omcom 4442
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-nul 3994  ax-pr 4069  ax-un 4293  ax-bd0 12592  ax-bdim 12593  ax-bdan 12594  ax-bdor 12595  ax-bdn 12596  ax-bdal 12597  ax-bdex 12598  ax-bdeq 12599  ax-bdel 12600  ax-bdsb 12601  ax-bdsep 12663  ax-infvn 12724
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-rab 2384  df-v 2643  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-sn 3480  df-pr 3481  df-uni 3684  df-int 3719  df-suc 4231  df-iom 4443  df-bdc 12620  df-bj-ind 12710
This theorem is referenced by:  bj-nn0suc  12747
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