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Theorem bj-nn0suc0 10462
Description: Constructive proof of a variant of nn0suc 4355. For a constructive proof of nn0suc 4355, see bj-nn0suc 10476. (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 2062 . . 3  |-  ( y  =  A  ->  (
y  =  (/)  <->  A  =  (/) ) )
2 eqeq1 2062 . . . 4  |-  ( y  =  A  ->  (
y  =  suc  x  <->  A  =  suc  x ) )
32rexeqbi1dv 2531 . . 3  |-  ( y  =  A  ->  ( E. x  e.  y 
y  =  suc  x  <->  E. x  e.  A  A  =  suc  x ) )
41, 3orbi12d 717 . 2  |-  ( y  =  A  ->  (
( y  =  (/)  \/ 
E. x  e.  y  y  =  suc  x
)  <->  ( A  =  (/)  \/  E. x  e.  A  A  =  suc  x ) ) )
5 tru 1263 . . 3  |- T.
6 a1tru 1275 . . . 4  |-  ( T. 
-> T.  )
76rgenw 2393 . . 3  |-  A. z  e.  om  ( T.  -> T.  )
8 bdeq0 10374 . . . . 5  |- BOUNDED  y  =  (/)
9 bdeqsuc 10388 . . . . . 6  |- BOUNDED  y  =  suc  x
109ax-bdex 10326 . . . . 5  |- BOUNDED  E. x  e.  y  y  =  suc  x
118, 10ax-bdor 10323 . . . 4  |- BOUNDED  ( y  =  (/)  \/ 
E. x  e.  y  y  =  suc  x
)
12 nfv 1437 . . . 4  |-  F/ y T.
13 orc 643 . . . . 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 1275 . . . . 5  |-  ( -.  ( y  =  z  ->  -.  ( y  =  (/)  \/  E. x  e.  y  y  =  suc  x ) )  -> T.  )
1615expi 577 . . . 4  |-  ( y  =  z  ->  (
( y  =  (/)  \/ 
E. x  e.  y  y  =  suc  x
)  -> T.  )
)
17 vex 2577 . . . . . . . . 9  |-  z  e. 
_V
1817sucid 4182 . . . . . . . 8  |-  z  e. 
suc  z
19 eleq2 2117 . . . . . . . 8  |-  ( y  =  suc  z  -> 
( z  e.  y  <-> 
z  e.  suc  z
) )
2018, 19mpbiri 161 . . . . . . 7  |-  ( y  =  suc  z  -> 
z  e.  y )
21 suceq 4167 . . . . . . . . 9  |-  ( x  =  z  ->  suc  x  =  suc  z )
2221eqeq2d 2067 . . . . . . . 8  |-  ( x  =  z  ->  (
y  =  suc  x  <->  y  =  suc  z ) )
2322rspcev 2673 . . . . . . 7  |-  ( ( z  e.  y  /\  y  =  suc  z )  ->  E. x  e.  y  y  =  suc  x
)
2420, 23mpancom 407 . . . . . 6  |-  ( y  =  suc  z  ->  E. x  e.  y 
y  =  suc  x
)
2524olcd 663 . . . . 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 10459 . . 3  |-  ( ( T.  /\  A. z  e.  om  ( T.  -> T.  ) )  ->  A. y  e.  om  ( y  =  (/)  \/  E. x  e.  y  y  =  suc  x ) )
285, 7, 27mp2an 410 . 2  |-  A. y  e.  om  ( y  =  (/)  \/  E. x  e.  y  y  =  suc  x )
294, 28vtoclri 2645 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 639    = wceq 1259   T. wtru 1260    e. wcel 1409   A.wral 2323   E.wrex 2324   (/)c0 3252   suc csuc 4130   omcom 4341
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-nul 3911  ax-pr 3972  ax-un 4198  ax-bd0 10320  ax-bdim 10321  ax-bdan 10322  ax-bdor 10323  ax-bdn 10324  ax-bdal 10325  ax-bdex 10326  ax-bdeq 10327  ax-bdel 10328  ax-bdsb 10329  ax-bdsep 10391  ax-infvn 10453
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-sn 3409  df-pr 3410  df-uni 3609  df-int 3644  df-suc 4136  df-iom 4342  df-bdc 10348  df-bj-ind 10438
This theorem is referenced by:  bj-nn0suc  10476
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