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Theorem elwina 8563
Description: Conditions of weak inaccessibility. (Contributed by Mario Carneiro, 22-Jun-2013.)
Assertion
Ref Expression
elwina  |-  ( A  e.  Inacc W  <->  ( A  =/=  (/)  /\  ( cf `  A )  =  A  /\  A. x  e.  A  E. y  e.  A  x  ~<  y
) )
Distinct variable group:    x, A, y

Proof of Theorem elwina
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 elex 2966 . 2  |-  ( A  e.  Inacc W  ->  A  e.  _V )
2 fvex 5744 . . . 4  |-  ( cf `  A )  e.  _V
3 eleq1 2498 . . . 4  |-  ( ( cf `  A )  =  A  ->  (
( cf `  A
)  e.  _V  <->  A  e.  _V ) )
42, 3mpbii 204 . . 3  |-  ( ( cf `  A )  =  A  ->  A  e.  _V )
543ad2ant2 980 . 2  |-  ( ( A  =/=  (/)  /\  ( cf `  A )  =  A  /\  A. x  e.  A  E. y  e.  A  x  ~<  y )  ->  A  e.  _V )
6 neeq1 2611 . . . 4  |-  ( z  =  A  ->  (
z  =/=  (/)  <->  A  =/=  (/) ) )
7 fveq2 5730 . . . . 5  |-  ( z  =  A  ->  ( cf `  z )  =  ( cf `  A
) )
8 eqeq12 2450 . . . . 5  |-  ( ( ( cf `  z
)  =  ( cf `  A )  /\  z  =  A )  ->  (
( cf `  z
)  =  z  <->  ( cf `  A )  =  A ) )
97, 8mpancom 652 . . . 4  |-  ( z  =  A  ->  (
( cf `  z
)  =  z  <->  ( cf `  A )  =  A ) )
10 rexeq 2907 . . . . 5  |-  ( z  =  A  ->  ( E. y  e.  z  x  ~<  y  <->  E. y  e.  A  x  ~<  y ) )
1110raleqbi1dv 2914 . . . 4  |-  ( z  =  A  ->  ( A. x  e.  z  E. y  e.  z  x  ~<  y  <->  A. x  e.  A  E. y  e.  A  x  ~<  y ) )
126, 9, 113anbi123d 1255 . . 3  |-  ( z  =  A  ->  (
( z  =/=  (/)  /\  ( cf `  z )  =  z  /\  A. x  e.  z  E. y  e.  z  x  ~<  y )  <->  ( A  =/=  (/)  /\  ( cf `  A
)  =  A  /\  A. x  e.  A  E. y  e.  A  x  ~<  y ) ) )
13 df-wina 8561 . . 3  |-  Inacc W  =  { z  |  ( z  =/=  (/)  /\  ( cf `  z )  =  z  /\  A. x  e.  z  E. y  e.  z  x  ~<  y ) }
1412, 13elab2g 3086 . 2  |-  ( A  e.  _V  ->  ( A  e.  Inacc W  <->  ( A  =/=  (/)  /\  ( cf `  A )  =  A  /\  A. x  e.  A  E. y  e.  A  x  ~<  y
) ) )
151, 5, 14pm5.21nii 344 1  |-  ( A  e.  Inacc W  <->  ( A  =/=  (/)  /\  ( cf `  A )  =  A  /\  A. x  e.  A  E. y  e.  A  x  ~<  y
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   E.wrex 2708   _Vcvv 2958   (/)c0 3630   class class class wbr 4214   ` cfv 5456    ~< csdm 7110   cfccf 7826   Inacc Wcwina 8559
This theorem is referenced by:  winaon  8565  inawina  8567  winacard  8569  winainf  8571  winalim2  8573  winafp  8574  gchina  8576
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-nul 4340
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-iota 5420  df-fv 5464  df-wina 8561
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