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Theorem elwina 8308
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 2796 . 2  |-  ( A  e.  Inacc W  ->  A  e.  _V )
2 fvex 5539 . . . 4  |-  ( cf `  A )  e.  _V
3 eleq1 2343 . . . 4  |-  ( ( cf `  A )  =  A  ->  (
( cf `  A
)  e.  _V  <->  A  e.  _V ) )
42, 3mpbii 202 . . 3  |-  ( ( cf `  A )  =  A  ->  A  e.  _V )
543ad2ant2 977 . 2  |-  ( ( A  =/=  (/)  /\  ( cf `  A )  =  A  /\  A. x  e.  A  E. y  e.  A  x  ~<  y )  ->  A  e.  _V )
6 neeq1 2454 . . . 4  |-  ( z  =  A  ->  (
z  =/=  (/)  <->  A  =/=  (/) ) )
7 fveq2 5525 . . . . 5  |-  ( z  =  A  ->  ( cf `  z )  =  ( cf `  A
) )
8 eqeq12 2295 . . . . 5  |-  ( ( ( cf `  z
)  =  ( cf `  A )  /\  z  =  A )  ->  (
( cf `  z
)  =  z  <->  ( cf `  A )  =  A ) )
97, 8mpancom 650 . . . 4  |-  ( z  =  A  ->  (
( cf `  z
)  =  z  <->  ( cf `  A )  =  A ) )
10 rexeq 2737 . . . . 5  |-  ( z  =  A  ->  ( E. y  e.  z  x  ~<  y  <->  E. y  e.  A  x  ~<  y ) )
1110raleqbi1dv 2744 . . . 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 1252 . . 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 8306 . . 3  |-  Inacc W  =  { z  |  ( z  =/=  (/)  /\  ( cf `  z )  =  z  /\  A. x  e.  z  E. y  e.  z  x  ~<  y ) }
1412, 13elab2g 2916 . 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 342 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 176    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   _Vcvv 2788   (/)c0 3455   class class class wbr 4023   ` cfv 5255    ~< csdm 6862   cfccf 7570   Inacc Wcwina 8304
This theorem is referenced by:  winaon  8310  inawina  8312  winacard  8314  winainf  8316  winalim2  8318  winafp  8319  gchina  8321
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-wina 8306
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