Prove irreflexivity of set membership in iset.mm about set.mm

[digama0]

Indeed, the proof follows from df-fr, or its proper class analogue ax-setind.

Suppose R Fr A, and B R B where B e. A. Let S = ( A \ { B } ). Suppose everything R-less than x is in S, and suppose x = B; then since B R B we have B e. S which is a contradiction. Thus x =/= B and hence x e. S. Therefore S is inductive, so A C_ S, but this is a contradiction since B e. A and B e/ S.

The exact same proof works in the proper class setting; we just take S = ( _V \ { B } ) and everything else is the same.

[digama0]

This proof also generalizes to show that there are no finite epsilon loops either (I think set.mm does the two and three cases), where you take S to be everything except the members of the loop.

[jkingdon]

elirr (irreflexivity) is merged into iset.mm and en2lop (lack of 2-cycle loops) is at #650 so I'm closing this.

The well-founded stuff remains at #646 .

[jkingdon]

en2lop (lack of 2-cycle loops) is at #650

In case anyone is still browsing old issues, this is a typo and en2lp is intended - http://us.metamath.org/ileuni/en2lp.html