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Theorem bj-inf2vn 10486
Description: A sufficient condition for  om to be a set. See bj-inf2vn2 10487 for the unbounded version from full set induction. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
bj-inf2vn.1  |- BOUNDED  A
Assertion
Ref Expression
bj-inf2vn  |-  ( A  e.  V  ->  ( A. x ( x  e.  A  <->  ( x  =  (/)  \/  E. y  e.  A  x  =  suc  y ) )  ->  A  =  om )
)
Distinct variable group:    x, y, A
Allowed substitution hints:    V( x, y)

Proof of Theorem bj-inf2vn
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 bj-inf2vnlem1 10482 . . 3  |-  ( A. x ( x  e.  A  <->  ( x  =  (/)  \/  E. y  e.  A  x  =  suc  y ) )  -> Ind  A )
2 bi1 115 . . . . . . 7  |-  ( ( x  e.  A  <->  ( x  =  (/)  \/  E. y  e.  A  x  =  suc  y ) )  -> 
( x  e.  A  ->  ( x  =  (/)  \/ 
E. y  e.  A  x  =  suc  y ) ) )
32alimi 1360 . . . . . 6  |-  ( A. x ( x  e.  A  <->  ( x  =  (/)  \/  E. y  e.  A  x  =  suc  y ) )  ->  A. x ( x  e.  A  ->  ( x  =  (/)  \/  E. y  e.  A  x  =  suc  y ) ) )
4 df-ral 2328 . . . . . 6  |-  ( A. x  e.  A  (
x  =  (/)  \/  E. y  e.  A  x  =  suc  y )  <->  A. x
( x  e.  A  ->  ( x  =  (/)  \/ 
E. y  e.  A  x  =  suc  y ) ) )
53, 4sylibr 141 . . . . 5  |-  ( A. x ( x  e.  A  <->  ( x  =  (/)  \/  E. y  e.  A  x  =  suc  y ) )  ->  A. x  e.  A  ( x  =  (/)  \/  E. y  e.  A  x  =  suc  y ) )
6 bj-inf2vn.1 . . . . . 6  |- BOUNDED  A
7 bdcv 10355 . . . . . 6  |- BOUNDED  z
86, 7bj-inf2vnlem3 10484 . . . . 5  |-  ( A. x  e.  A  (
x  =  (/)  \/  E. y  e.  A  x  =  suc  y )  -> 
(Ind  z  ->  A  C_  z ) )
95, 8syl 14 . . . 4  |-  ( A. x ( x  e.  A  <->  ( x  =  (/)  \/  E. y  e.  A  x  =  suc  y ) )  -> 
(Ind  z  ->  A  C_  z ) )
109alrimiv 1770 . . 3  |-  ( A. x ( x  e.  A  <->  ( x  =  (/)  \/  E. y  e.  A  x  =  suc  y ) )  ->  A. z (Ind  z  ->  A  C_  z ) )
111, 10jca 294 . 2  |-  ( A. x ( x  e.  A  <->  ( x  =  (/)  \/  E. y  e.  A  x  =  suc  y ) )  -> 
(Ind  A  /\  A. z (Ind  z  ->  A 
C_  z ) ) )
12 bj-om 10448 . 2  |-  ( A  e.  V  ->  ( A  =  om  <->  (Ind  A  /\  A. z (Ind  z  ->  A  C_  z
) ) ) )
1311, 12syl5ibr 149 1  |-  ( A  e.  V  ->  ( A. x ( x  e.  A  <->  ( x  =  (/)  \/  E. y  e.  A  x  =  suc  y ) )  ->  A  =  om )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    \/ wo 639   A.wal 1257    = wceq 1259    e. wcel 1409   A.wral 2323   E.wrex 2324    C_ wss 2945   (/)c0 3252   suc csuc 4130   omcom 4341  BOUNDED wbdc 10347  Ind wind 10437
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-bdor 10323  ax-bdex 10326  ax-bdeq 10327  ax-bdel 10328  ax-bdsb 10329  ax-bdsep 10391  ax-bdsetind 10480
This theorem depends on definitions:  df-bi 114  df-tru 1262  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-omex2  10489  bj-nn0sucALT  10490
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