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Theorem elwina 9053
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 3115 . 2  |-  ( A  e.  InaccW  ->  A  e.  _V )
2 fvex 5858 . . . 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 1016 . 2  |-  ( ( A  =/=  (/)  /\  ( cf `  A )  =  A  /\  A. x  e.  A  E. y  e.  A  x  ~<  y )  ->  A  e.  _V )
6 neeq1 2735 . . . 4  |-  ( z  =  A  ->  (
z  =/=  (/)  <->  A  =/=  (/) ) )
7 fveq2 5848 . . . . 5  |-  ( z  =  A  ->  ( cf `  z )  =  ( cf `  A
) )
8 eqeq12 2473 . . . . 5  |-  ( ( ( cf `  z
)  =  ( cf `  A )  /\  z  =  A )  ->  (
( cf `  z
)  =  z  <->  ( cf `  A )  =  A ) )
97, 8mpancom 667 . . . 4  |-  ( z  =  A  ->  (
( cf `  z
)  =  z  <->  ( cf `  A )  =  A ) )
10 rexeq 3052 . . . . 5  |-  ( z  =  A  ->  ( E. y  e.  z  x  ~<  y  <->  E. y  e.  A  x  ~<  y ) )
1110raleqbi1dv 3059 . . . 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 1297 . . 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 9051 . . 3  |-  InaccW  =  { z  |  ( z  =/=  (/)  /\  ( cf `  z )  =  z  /\  A. x  e.  z  E. y  e.  z  x  ~<  y ) }
1412, 13elab2g 3245 . 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 351 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 184    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   E.wrex 2805   _Vcvv 3106   (/)c0 3783   class class class wbr 4439   ` cfv 5570    ~< csdm 7508   cfccf 8309   InaccWcwina 9049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-nul 4568
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-iota 5534  df-fv 5578  df-wina 9051
This theorem is referenced by:  winaon  9055  inawina  9057  winacard  9059  winainf  9061  winalim2  9063  winafp  9064  gchina  9066
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