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Theorem elwina 9111
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 3090 . 2  |-  ( A  e.  InaccW  ->  A  e.  _V )
2 fvex 5887 . . . 4  |-  ( cf `  A )  e.  _V
3 eleq1 2494 . . . 4  |-  ( ( cf `  A )  =  A  ->  (
( cf `  A
)  e.  _V  <->  A  e.  _V ) )
42, 3mpbii 214 . . 3  |-  ( ( cf `  A )  =  A  ->  A  e.  _V )
543ad2ant2 1027 . 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 5877 . . . . 5  |-  ( z  =  A  ->  ( cf `  z )  =  ( cf `  A
) )
8 eqeq12 2441 . . . . 5  |-  ( ( ( cf `  z
)  =  ( cf `  A )  /\  z  =  A )  ->  (
( cf `  z
)  =  z  <->  ( cf `  A )  =  A ) )
97, 8mpancom 673 . . . 4  |-  ( z  =  A  ->  (
( cf `  z
)  =  z  <->  ( cf `  A )  =  A ) )
10 rexeq 3026 . . . . 5  |-  ( z  =  A  ->  ( E. y  e.  z  x  ~<  y  <->  E. y  e.  A  x  ~<  y ) )
1110raleqbi1dv 3033 . . . 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 1335 . . 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 9109 . . 3  |-  InaccW  =  { z  |  ( z  =/=  (/)  /\  ( cf `  z )  =  z  /\  A. x  e.  z  E. y  e.  z  x  ~<  y ) }
1412, 13elab2g 3220 . 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 354 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 187    /\ w3a 982    = wceq 1437    e. wcel 1868    =/= wne 2618   A.wral 2775   E.wrex 2776   _Vcvv 3081   (/)c0 3761   class class class wbr 4420   ` cfv 5597    ~< csdm 7572   cfccf 8372   InaccWcwina 9107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-nul 4551
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-br 4421  df-iota 5561  df-fv 5605  df-wina 9109
This theorem is referenced by:  winaon  9113  inawina  9115  winacard  9117  winainf  9119  winalim2  9121  winafp  9122  gchina  9124
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