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Theorem bj-nn0suc0 13207
Description: Constructive proof of a variant of nn0suc 4518. For a constructive proof of nn0suc 4518, see bj-nn0suc 13221. (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 2146 . . 3  |-  ( y  =  A  ->  (
y  =  (/)  <->  A  =  (/) ) )
2 eqeq1 2146 . . . 4  |-  ( y  =  A  ->  (
y  =  suc  x  <->  A  =  suc  x ) )
32rexeqbi1dv 2635 . . 3  |-  ( y  =  A  ->  ( E. x  e.  y 
y  =  suc  x  <->  E. x  e.  A  A  =  suc  x ) )
41, 3orbi12d 782 . 2  |-  ( y  =  A  ->  (
( y  =  (/)  \/ 
E. x  e.  y  y  =  suc  x
)  <->  ( A  =  (/)  \/  E. x  e.  A  A  =  suc  x ) ) )
5 tru 1335 . . 3  |- T.
6 a1tru 1347 . . . 4  |-  ( T. 
-> T.  )
76rgenw 2487 . . 3  |-  A. z  e.  om  ( T.  -> T.  )
8 bdeq0 13124 . . . . 5  |- BOUNDED  y  =  (/)
9 bdeqsuc 13138 . . . . . 6  |- BOUNDED  y  =  suc  x
109ax-bdex 13076 . . . . 5  |- BOUNDED  E. x  e.  y  y  =  suc  x
118, 10ax-bdor 13073 . . . 4  |- BOUNDED  ( y  =  (/)  \/ 
E. x  e.  y  y  =  suc  x
)
12 nfv 1508 . . . 4  |-  F/ y T.
13 orc 701 . . . . 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 1347 . . . . 5  |-  ( -.  ( y  =  z  ->  -.  ( y  =  (/)  \/  E. x  e.  y  y  =  suc  x ) )  -> T.  )
1615expi 627 . . . 4  |-  ( y  =  z  ->  (
( y  =  (/)  \/ 
E. x  e.  y  y  =  suc  x
)  -> T.  )
)
17 vex 2689 . . . . . . . . 9  |-  z  e. 
_V
1817sucid 4339 . . . . . . . 8  |-  z  e. 
suc  z
19 eleq2 2203 . . . . . . . 8  |-  ( y  =  suc  z  -> 
( z  e.  y  <-> 
z  e.  suc  z
) )
2018, 19mpbiri 167 . . . . . . 7  |-  ( y  =  suc  z  -> 
z  e.  y )
21 suceq 4324 . . . . . . . . 9  |-  ( x  =  z  ->  suc  x  =  suc  z )
2221eqeq2d 2151 . . . . . . . 8  |-  ( x  =  z  ->  (
y  =  suc  x  <->  y  =  suc  z ) )
2322rspcev 2789 . . . . . . 7  |-  ( ( z  e.  y  /\  y  =  suc  z )  ->  E. x  e.  y  y  =  suc  x
)
2420, 23mpancom 418 . . . . . 6  |-  ( y  =  suc  z  ->  E. x  e.  y 
y  =  suc  x
)
2524olcd 723 . . . . 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 13204 . . 3  |-  ( ( T.  /\  A. z  e.  om  ( T.  -> T.  ) )  ->  A. y  e.  om  ( y  =  (/)  \/  E. x  e.  y  y  =  suc  x ) )
285, 7, 27mp2an 422 . 2  |-  A. y  e.  om  ( y  =  (/)  \/  E. x  e.  y  y  =  suc  x )
294, 28vtoclri 2761 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 697    = wceq 1331   T. wtru 1332    e. wcel 1480   A.wral 2416   E.wrex 2417   (/)c0 3363   suc csuc 4287   omcom 4504
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-nul 4054  ax-pr 4131  ax-un 4355  ax-bd0 13070  ax-bdim 13071  ax-bdan 13072  ax-bdor 13073  ax-bdn 13074  ax-bdal 13075  ax-bdex 13076  ax-bdeq 13077  ax-bdel 13078  ax-bdsb 13079  ax-bdsep 13141  ax-infvn 13198
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-sn 3533  df-pr 3534  df-uni 3737  df-int 3772  df-suc 4293  df-iom 4505  df-bdc 13098  df-bj-ind 13184
This theorem is referenced by:  bj-nn0suc  13221
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