This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem ordirr

Description: No ordinal class is a member of itself. In other words, the membership relation is irreflexive on ordinal classes. Theorem 2.2(i) of BellMachover p. 469, generalized to classes. Theorem 1.9(i) of Schloeder p. 1. We prove this without invoking the Axiom of Regularity. (Contributed by NM, 2-Jan-1994)

Ref Expression
Assertion ordirr 
|- ( Ord A -> -. A e. A )

Proof

Step Hyp Ref Expression
1 ordfr 
 |-  ( Ord A -> _E Fr A )
2 efrirr 
 |-  ( _E Fr A -> -. A e. A )
3 1 2 syl 
 |-  ( Ord A -> -. A e. A )