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Theorem elwina 9129
Description: Conditions of weak inaccessibility. (Contributed by Mario Carneiro, 22-Jun-2013.)
Assertion
Ref Expression
elwina  |-  ( A  e.  InaccW  <->  ( 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 3040 . 2  |-  ( A  e.  InaccW  ->  A  e.  _V )
2 fvex 5889 . . . 4  |-  ( cf `  A )  e.  _V
3 eleq1 2537 . . . 4  |-  ( ( cf `  A )  =  A  ->  (
( cf `  A
)  e.  _V  <->  A  e.  _V ) )
42, 3mpbii 216 . . 3  |-  ( ( cf `  A )  =  A  ->  A  e.  _V )
543ad2ant2 1052 . 2  |-  ( ( A  =/=  (/)  /\  ( cf `  A )  =  A  /\  A. x  e.  A  E. y  e.  A  x  ~<  y )  ->  A  e.  _V )
6 neeq1 2705 . . . 4  |-  ( z  =  A  ->  (
z  =/=  (/)  <->  A  =/=  (/) ) )
7 fveq2 5879 . . . . 5  |-  ( z  =  A  ->  ( cf `  z )  =  ( cf `  A
) )
8 eqeq12 2484 . . . . 5  |-  ( ( ( cf `  z
)  =  ( cf `  A )  /\  z  =  A )  ->  (
( cf `  z
)  =  z  <->  ( cf `  A )  =  A ) )
97, 8mpancom 682 . . . 4  |-  ( z  =  A  ->  (
( cf `  z
)  =  z  <->  ( cf `  A )  =  A ) )
10 rexeq 2974 . . . . 5  |-  ( z  =  A  ->  ( E. y  e.  z  x  ~<  y  <->  E. y  e.  A  x  ~<  y ) )
1110raleqbi1dv 2981 . . . 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 1365 . . 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 9127 . . 3  |-  InaccW  =  { z  |  ( z  =/=  (/)  /\  ( cf `  z )  =  z  /\  A. x  e.  z  E. y  e.  z  x  ~<  y ) }
1412, 13elab2g 3175 . 2  |-  ( A  e.  _V  ->  ( A  e.  InaccW  <->  ( A  =/=  (/)  /\  ( cf `  A )  =  A  /\  A. x  e.  A  E. y  e.  A  x  ~<  y
) ) )
151, 5, 14pm5.21nii 360 1  |-  ( A  e.  InaccW  <->  ( A  =/=  (/)  /\  ( cf `  A )  =  A  /\  A. x  e.  A  E. y  e.  A  x  ~<  y
) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   A.wral 2756   E.wrex 2757   _Vcvv 3031   (/)c0 3722   class class class wbr 4395   ` cfv 5589    ~< csdm 7586   cfccf 8389   InaccWcwina 9125
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-nul 4527
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-iota 5553  df-fv 5597  df-wina 9127
This theorem is referenced by:  winaon  9131  inawina  9133  winacard  9135  winainf  9137  winalim2  9139  winafp  9140  gchina  9142
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