Comments (4)
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.
from set.mm.
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.
from set.mm.
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 .
from set.mm.
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
from set.mm.
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from set.mm.