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Theorem bj-nn0suc0 14973
Description: Constructive proof of a variant of nn0suc 4615. For a constructive proof of nn0suc 4615, see bj-nn0suc 14987. (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 2194 . . 3  |-  ( y  =  A  ->  (
y  =  (/)  <->  A  =  (/) ) )
2 eqeq1 2194 . . . 4  |-  ( y  =  A  ->  (
y  =  suc  x  <->  A  =  suc  x ) )
32rexeqbi1dv 2692 . . 3  |-  ( y  =  A  ->  ( E. x  e.  y 
y  =  suc  x  <->  E. x  e.  A  A  =  suc  x ) )
41, 3orbi12d 794 . 2  |-  ( y  =  A  ->  (
( y  =  (/)  \/ 
E. x  e.  y  y  =  suc  x
)  <->  ( A  =  (/)  \/  E. x  e.  A  A  =  suc  x ) ) )
5 tru 1367 . . 3  |- T.
6 trud 1379 . . . 4  |-  ( T. 
-> T.  )
76rgenw 2542 . . 3  |-  A. z  e.  om  ( T.  -> T.  )
8 bdeq0 14890 . . . . 5  |- BOUNDED  y  =  (/)
9 bdeqsuc 14904 . . . . . 6  |- BOUNDED  y  =  suc  x
109ax-bdex 14842 . . . . 5  |- BOUNDED  E. x  e.  y  y  =  suc  x
118, 10ax-bdor 14839 . . . 4  |- BOUNDED  ( y  =  (/)  \/ 
E. x  e.  y  y  =  suc  x
)
12 nfv 1538 . . . 4  |-  F/ y T.
13 orc 713 . . . . 5  |-  ( y  =  (/)  ->  ( y  =  (/)  \/  E. x  e.  y  y  =  suc  x ) )
1413a1d 22 . . . 4  |-  ( y  =  (/)  ->  ( T. 
->  ( y  =  (/)  \/ 
E. x  e.  y  y  =  suc  x
) ) )
15 trud 1379 . . . . 5  |-  ( -.  ( y  =  z  ->  -.  ( y  =  (/)  \/  E. x  e.  y  y  =  suc  x ) )  -> T.  )
1615expi 639 . . . 4  |-  ( y  =  z  ->  (
( y  =  (/)  \/ 
E. x  e.  y  y  =  suc  x
)  -> T.  )
)
17 vex 2752 . . . . . . . . 9  |-  z  e. 
_V
1817sucid 4429 . . . . . . . 8  |-  z  e. 
suc  z
19 eleq2 2251 . . . . . . . 8  |-  ( y  =  suc  z  -> 
( z  e.  y  <-> 
z  e.  suc  z
) )
2018, 19mpbiri 168 . . . . . . 7  |-  ( y  =  suc  z  -> 
z  e.  y )
21 suceq 4414 . . . . . . . . 9  |-  ( x  =  z  ->  suc  x  =  suc  z )
2221eqeq2d 2199 . . . . . . . 8  |-  ( x  =  z  ->  (
y  =  suc  x  <->  y  =  suc  z ) )
2322rspcev 2853 . . . . . . 7  |-  ( ( z  e.  y  /\  y  =  suc  z )  ->  E. x  e.  y  y  =  suc  x
)
2420, 23mpancom 422 . . . . . 6  |-  ( y  =  suc  z  ->  E. x  e.  y 
y  =  suc  x
)
2524olcd 735 . . . . 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 14970 . . 3  |-  ( ( T.  /\  A. z  e.  om  ( T.  -> T.  ) )  ->  A. y  e.  om  ( y  =  (/)  \/  E. x  e.  y  y  =  suc  x ) )
285, 7, 27mp2an 426 . 2  |-  A. y  e.  om  ( y  =  (/)  \/  E. x  e.  y  y  =  suc  x )
294, 28vtoclri 2824 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 709    = wceq 1363   T. wtru 1364    e. wcel 2158   A.wral 2465   E.wrex 2466   (/)c0 3434   suc csuc 4377   omcom 4601
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-nul 4141  ax-pr 4221  ax-un 4445  ax-bd0 14836  ax-bdim 14837  ax-bdan 14838  ax-bdor 14839  ax-bdn 14840  ax-bdal 14841  ax-bdex 14842  ax-bdeq 14843  ax-bdel 14844  ax-bdsb 14845  ax-bdsep 14907  ax-infvn 14964
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-sn 3610  df-pr 3611  df-uni 3822  df-int 3857  df-suc 4383  df-iom 4602  df-bdc 14864  df-bj-ind 14950
This theorem is referenced by:  bj-nn0suc  14987
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