Metamath Proof Explorer


Theorem negsfo

Description: Function statement for surreal negation. (Contributed by Scott Fenton, 3-Feb-2025)

Ref Expression
Assertion negsfo Could not format assertion : No typesetting found for |- -us : No -onto-> No with typecode |-

Proof

Step Hyp Ref Expression
1 negsf Could not format -us : No --> No : No typesetting found for |- -us : No --> No with typecode |-
2 negscl Could not format ( x e. No -> ( -us ` x ) e. No ) : No typesetting found for |- ( x e. No -> ( -us ` x ) e. No ) with typecode |-
3 negnegs Could not format ( x e. No -> ( -us ` ( -us ` x ) ) = x ) : No typesetting found for |- ( x e. No -> ( -us ` ( -us ` x ) ) = x ) with typecode |-
4 3 eqcomd Could not format ( x e. No -> x = ( -us ` ( -us ` x ) ) ) : No typesetting found for |- ( x e. No -> x = ( -us ` ( -us ` x ) ) ) with typecode |-
5 fveq2 Could not format ( y = ( -us ` x ) -> ( -us ` y ) = ( -us ` ( -us ` x ) ) ) : No typesetting found for |- ( y = ( -us ` x ) -> ( -us ` y ) = ( -us ` ( -us ` x ) ) ) with typecode |-
6 5 eqeq2d Could not format ( y = ( -us ` x ) -> ( x = ( -us ` y ) <-> x = ( -us ` ( -us ` x ) ) ) ) : No typesetting found for |- ( y = ( -us ` x ) -> ( x = ( -us ` y ) <-> x = ( -us ` ( -us ` x ) ) ) ) with typecode |-
7 6 rspcev Could not format ( ( ( -us ` x ) e. No /\ x = ( -us ` ( -us ` x ) ) ) -> E. y e. No x = ( -us ` y ) ) : No typesetting found for |- ( ( ( -us ` x ) e. No /\ x = ( -us ` ( -us ` x ) ) ) -> E. y e. No x = ( -us ` y ) ) with typecode |-
8 2 4 7 syl2anc Could not format ( x e. No -> E. y e. No x = ( -us ` y ) ) : No typesetting found for |- ( x e. No -> E. y e. No x = ( -us ` y ) ) with typecode |-
9 8 rgen Could not format A. x e. No E. y e. No x = ( -us ` y ) : No typesetting found for |- A. x e. No E. y e. No x = ( -us ` y ) with typecode |-
10 dffo3 Could not format ( -us : No -onto-> No <-> ( -us : No --> No /\ A. x e. No E. y e. No x = ( -us ` y ) ) ) : No typesetting found for |- ( -us : No -onto-> No <-> ( -us : No --> No /\ A. x e. No E. y e. No x = ( -us ` y ) ) ) with typecode |-
11 1 9 10 mpbir2an Could not format -us : No -onto-> No : No typesetting found for |- -us : No -onto-> No with typecode |-