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Theorem elina 9112
Description: Conditions of strong inaccessibility. (Contributed by Mario Carneiro, 22-Jun-2013.)
Assertion
Ref Expression
elina  |-  ( A  e.  Inacc 
<->  ( A  =/=  (/)  /\  ( cf `  A )  =  A  /\  A. x  e.  A  ~P x  ~<  A ) )
Distinct variable group:    x, A

Proof of Theorem elina
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 elex 3054 . 2  |-  ( A  e.  Inacc  ->  A  e.  _V )
2 fvex 5875 . . . 4  |-  ( cf `  A )  e.  _V
3 eleq1 2517 . . . 4  |-  ( ( cf `  A )  =  A  ->  (
( cf `  A
)  e.  _V  <->  A  e.  _V ) )
42, 3mpbii 215 . . 3  |-  ( ( cf `  A )  =  A  ->  A  e.  _V )
543ad2ant2 1030 . 2  |-  ( ( A  =/=  (/)  /\  ( cf `  A )  =  A  /\  A. x  e.  A  ~P x  ~<  A )  ->  A  e.  _V )
6 neeq1 2686 . . . 4  |-  ( y  =  A  ->  (
y  =/=  (/)  <->  A  =/=  (/) ) )
7 fveq2 5865 . . . . 5  |-  ( y  =  A  ->  ( cf `  y )  =  ( cf `  A
) )
8 eqeq12 2464 . . . . 5  |-  ( ( ( cf `  y
)  =  ( cf `  A )  /\  y  =  A )  ->  (
( cf `  y
)  =  y  <->  ( cf `  A )  =  A ) )
97, 8mpancom 675 . . . 4  |-  ( y  =  A  ->  (
( cf `  y
)  =  y  <->  ( cf `  A )  =  A ) )
10 breq2 4406 . . . . 5  |-  ( y  =  A  ->  ( ~P x  ~<  y  <->  ~P x  ~<  A ) )
1110raleqbi1dv 2995 . . . 4  |-  ( y  =  A  ->  ( A. x  e.  y  ~P x  ~<  y  <->  A. x  e.  A  ~P x  ~<  A ) )
126, 9, 113anbi123d 1339 . . 3  |-  ( y  =  A  ->  (
( y  =/=  (/)  /\  ( cf `  y )  =  y  /\  A. x  e.  y  ~P x  ~<  y )  <->  ( A  =/=  (/)  /\  ( cf `  A )  =  A  /\  A. x  e.  A  ~P x  ~<  A ) ) )
13 df-ina 9110 . . 3  |-  Inacc  =  {
y  |  ( y  =/=  (/)  /\  ( cf `  y )  =  y  /\  A. x  e.  y  ~P x  ~<  y ) }
1412, 13elab2g 3187 . 2  |-  ( A  e.  _V  ->  ( A  e.  Inacc  <->  ( A  =/=  (/)  /\  ( cf `  A )  =  A  /\  A. x  e.  A  ~P x  ~<  A ) ) )
151, 5, 14pm5.21nii 355 1  |-  ( A  e.  Inacc 
<->  ( A  =/=  (/)  /\  ( cf `  A )  =  A  /\  A. x  e.  A  ~P x  ~<  A ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188    /\ w3a 985    = wceq 1444    e. wcel 1887    =/= wne 2622   A.wral 2737   _Vcvv 3045   (/)c0 3731   ~Pcpw 3951   class class class wbr 4402   ` cfv 5582    ~< csdm 7568   cfccf 8371   Inacccina 9108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-nul 4534
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-iota 5546  df-fv 5590  df-ina 9110
This theorem is referenced by:  inawina  9115  omina  9116  gchina  9124  inar1  9200  inatsk  9203  tskcard  9206  tskuni  9208  gruina  9243  grur1  9245
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