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Theorem elina 8968
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 3087 . 2  |-  ( A  e.  Inacc  ->  A  e.  _V )
2 fvex 5812 . . . 4  |-  ( cf `  A )  e.  _V
3 eleq1 2526 . . . 4  |-  ( ( cf `  A )  =  A  ->  (
( cf `  A
)  e.  _V  <->  A  e.  _V ) )
42, 3mpbii 211 . . 3  |-  ( ( cf `  A )  =  A  ->  A  e.  _V )
543ad2ant2 1010 . 2  |-  ( ( A  =/=  (/)  /\  ( cf `  A )  =  A  /\  A. x  e.  A  ~P x  ~<  A )  ->  A  e.  _V )
6 neeq1 2733 . . . 4  |-  ( y  =  A  ->  (
y  =/=  (/)  <->  A  =/=  (/) ) )
7 fveq2 5802 . . . . 5  |-  ( y  =  A  ->  ( cf `  y )  =  ( cf `  A
) )
8 eqeq12 2473 . . . . 5  |-  ( ( ( cf `  y
)  =  ( cf `  A )  /\  y  =  A )  ->  (
( cf `  y
)  =  y  <->  ( cf `  A )  =  A ) )
97, 8mpancom 669 . . . 4  |-  ( y  =  A  ->  (
( cf `  y
)  =  y  <->  ( cf `  A )  =  A ) )
10 breq2 4407 . . . . 5  |-  ( y  =  A  ->  ( ~P x  ~<  y  <->  ~P x  ~<  A ) )
1110raleqbi1dv 3031 . . . 4  |-  ( y  =  A  ->  ( A. x  e.  y  ~P x  ~<  y  <->  A. x  e.  A  ~P x  ~<  A ) )
126, 9, 113anbi123d 1290 . . 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 8966 . . 3  |-  Inacc  =  {
y  |  ( y  =/=  (/)  /\  ( cf `  y )  =  y  /\  A. x  e.  y  ~P x  ~<  y ) }
1412, 13elab2g 3215 . 2  |-  ( A  e.  _V  ->  ( A  e.  Inacc  <->  ( A  =/=  (/)  /\  ( cf `  A )  =  A  /\  A. x  e.  A  ~P x  ~<  A ) ) )
151, 5, 14pm5.21nii 353 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 965    = wceq 1370    e. wcel 1758    =/= wne 2648   A.wral 2799   _Vcvv 3078   (/)c0 3748   ~Pcpw 3971   class class class wbr 4403   ` cfv 5529    ~< csdm 7422   cfccf 8221   Inacccina 8964
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-nul 4532
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-iota 5492  df-fv 5537  df-ina 8966
This theorem is referenced by:  inawina  8971  omina  8972  gchina  8980  inar1  9056  inatsk  9059  tskcard  9062  tskuni  9064  gruina  9099  grur1  9101
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