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Theorem bj-inf2vn 14009
Description: A sufficient condition for  om to be a set. See bj-inf2vn2 14010 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 14005 . . 3  |-  ( A. x ( x  e.  A  <->  ( x  =  (/)  \/  E. y  e.  A  x  =  suc  y ) )  -> Ind  A )
2 biimp 117 . . . . . . 7  |-  ( ( x  e.  A  <->  ( x  =  (/)  \/  E. y  e.  A  x  =  suc  y ) )  -> 
( x  e.  A  ->  ( x  =  (/)  \/ 
E. y  e.  A  x  =  suc  y ) ) )
32alimi 1448 . . . . . 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 2453 . . . . . 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 133 . . . . 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 13883 . . . . . 6  |- BOUNDED  z
86, 7bj-inf2vnlem3 14007 . . . . 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 1867 . . 3  |-  ( A. x ( x  e.  A  <->  ( x  =  (/)  \/  E. y  e.  A  x  =  suc  y ) )  ->  A. z (Ind  z  ->  A  C_  z ) )
111, 10jca 304 . 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 13972 . 2  |-  ( A  e.  V  ->  ( A  =  om  <->  (Ind  A  /\  A. z (Ind  z  ->  A  C_  z
) ) ) )
1311, 12syl5ibr 155 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 103    <-> wb 104    \/ wo 703   A.wal 1346    = wceq 1348    e. wcel 2141   A.wral 2448   E.wrex 2449    C_ wss 3121   (/)c0 3414   suc csuc 4350   omcom 4574  BOUNDED wbdc 13875  Ind wind 13961
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-nul 4115  ax-pr 4194  ax-un 4418  ax-bd0 13848  ax-bdim 13849  ax-bdor 13851  ax-bdex 13854  ax-bdeq 13855  ax-bdel 13856  ax-bdsb 13857  ax-bdsep 13919  ax-bdsetind 14003
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-sn 3589  df-pr 3590  df-uni 3797  df-int 3832  df-suc 4356  df-iom 4575  df-bdc 13876  df-bj-ind 13962
This theorem is referenced by:  bj-omex2  14012  bj-nn0sucALT  14013
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