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Theorem elina 9094
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 3067 . 2  |-  ( A  e.  Inacc  ->  A  e.  _V )
2 fvex 5858 . . . 4  |-  ( cf `  A )  e.  _V
3 eleq1 2474 . . . 4  |-  ( ( cf `  A )  =  A  ->  (
( cf `  A
)  e.  _V  <->  A  e.  _V ) )
42, 3mpbii 211 . . 3  |-  ( ( cf `  A )  =  A  ->  A  e.  _V )
543ad2ant2 1019 . 2  |-  ( ( A  =/=  (/)  /\  ( cf `  A )  =  A  /\  A. x  e.  A  ~P x  ~<  A )  ->  A  e.  _V )
6 neeq1 2684 . . . 4  |-  ( y  =  A  ->  (
y  =/=  (/)  <->  A  =/=  (/) ) )
7 fveq2 5848 . . . . 5  |-  ( y  =  A  ->  ( cf `  y )  =  ( cf `  A
) )
8 eqeq12 2421 . . . . 5  |-  ( ( ( cf `  y
)  =  ( cf `  A )  /\  y  =  A )  ->  (
( cf `  y
)  =  y  <->  ( cf `  A )  =  A ) )
97, 8mpancom 667 . . . 4  |-  ( y  =  A  ->  (
( cf `  y
)  =  y  <->  ( cf `  A )  =  A ) )
10 breq2 4398 . . . . 5  |-  ( y  =  A  ->  ( ~P x  ~<  y  <->  ~P x  ~<  A ) )
1110raleqbi1dv 3011 . . . 4  |-  ( y  =  A  ->  ( A. x  e.  y  ~P x  ~<  y  <->  A. x  e.  A  ~P x  ~<  A ) )
126, 9, 113anbi123d 1301 . . 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 9092 . . 3  |-  Inacc  =  {
y  |  ( y  =/=  (/)  /\  ( cf `  y )  =  y  /\  A. x  e.  y  ~P x  ~<  y ) }
1412, 13elab2g 3197 . 2  |-  ( A  e.  _V  ->  ( A  e.  Inacc  <->  ( A  =/=  (/)  /\  ( cf `  A )  =  A  /\  A. x  e.  A  ~P x  ~<  A ) ) )
151, 5, 14pm5.21nii 351 1  |-  ( A  e.  Inacc 
<->  ( A  =/=  (/)  /\  ( cf `  A )  =  A  /\  A. x  e.  A  ~P x  ~<  A ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   A.wral 2753   _Vcvv 3058   (/)c0 3737   ~Pcpw 3954   class class class wbr 4394   ` cfv 5568    ~< csdm 7552   cfccf 8349   Inacccina 9090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-nul 4524
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-iota 5532  df-fv 5576  df-ina 9092
This theorem is referenced by:  inawina  9097  omina  9098  gchina  9106  inar1  9182  inatsk  9185  tskcard  9188  tskuni  9190  gruina  9225  grur1  9227
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