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. We prove this without invoking the Axiom of Regularity. (Contributed by NM, 2-Jan-1994)

Ref Expression
Assertion ordirr Ord A ¬ A A


Step Hyp Ref Expression
1 ordfr Ord A E Fr A
2 efrirr E Fr A ¬ A A
3 1 2 syl Ord A ¬ A A