Metamath Proof Explorer


Theorem bnj69

Description: Existence of a minimal element in certain classes: if R is well-founded and set-like on A , then every nonempty subclass of A has a minimal element. The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf . (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

Ref Expression
Assertion bnj69 RFrSeABABxByB¬yRx

Proof

Step Hyp Ref Expression
1 biid RFrSeABABRFrSeABAB
2 biid xByByRxxByByRx
3 biid yB¬yRxyB¬yRx
4 1 2 3 bnj1189 RFrSeABABxByB¬yRx