Metamath Proof Explorer

Theorem elwina

Description: Conditions of weak inaccessibility. (Contributed by Mario Carneiro, 22-Jun-2013)

Ref Expression
Assertion elwina 
|- ( A e. InaccW <-> ( A =/= (/) /\ ( cf ` A ) = A /\ A. x e. A E. y e. A x ~< y ) )

Proof

Step Hyp Ref Expression
1 elex 
 |-  ( A e. InaccW -> A e. _V )
2 fvex 
 |-  ( cf ` A ) e. _V
3 eleq1 
 |-  ( ( cf ` A ) = A -> ( ( cf ` A ) e. _V <-> A e. _V ) )
4 2 3 mpbii 
 |-  ( ( cf ` A ) = A -> A e. _V )
5 4 3ad2ant2 
 |-  ( ( A =/= (/) /\ ( cf ` A ) = A /\ A. x e. A E. y e. A x ~< y ) -> A e. _V )
6 neeq1 
 |-  ( z = A -> ( z =/= (/) <-> A =/= (/) ) )
7 fveq2 
 |-  ( z = A -> ( cf ` z ) = ( cf ` A ) )
8 eqeq12 
 |-  ( ( ( cf ` z ) = ( cf ` A ) /\ z = A ) -> ( ( cf ` z ) = z <-> ( cf ` A ) = A ) )
9 7 8 mpancom 
 |-  ( z = A -> ( ( cf ` z ) = z <-> ( cf ` A ) = A ) )
10 rexeq 
 |-  ( z = A -> ( E. y e. z x ~< y <-> E. y e. A x ~< y ) )
11 10 raleqbi1dv 
 |-  ( z = A -> ( A. x e. z E. y e. z x ~< y <-> A. x e. A E. y e. A x ~< y ) )
12 6 9 11 3anbi123d 
 |-  ( 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 
 |-  InaccW = { z | ( z =/= (/) /\ ( cf ` z ) = z /\ A. x e. z E. y e. z x ~< y ) }
14 12 13 elab2g 
 |-  ( A e. _V -> ( A e. InaccW <-> ( A =/= (/) /\ ( cf ` A ) = A /\ A. x e. A E. y e. A x ~< y ) ) )
15 1 5 14 pm5.21nii 
 |-  ( A e. InaccW <-> ( A =/= (/) /\ ( cf ` A ) = A /\ A. x e. A E. y e. A x ~< y ) )