Metamath Proof Explorer

Theorem bnj1228

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
Hypothesis bnj1228.1 ⊢w∈B→∀xw∈B
Assertion bnj1228 ⊢RFrSeA∧B⊆A∧B≠∅→∃x∈B∀y∈B¬yRx

Proof

Step Hyp Ref Expression
1 bnj1228.1 ⊢w∈B→∀xw∈B
2 bnj69 ⊢RFrSeA∧B⊆A∧B≠∅→∃z∈B∀y∈B¬yRz
3 nfv ⊢Ⅎzx∈B∧∀y∈B¬yRx
4 1 nfcii ⊢Ⅎ_xB
5 4 nfcri ⊢Ⅎxz∈B
6 nfv ⊢Ⅎx¬yRz
7 4 6 nfralw ⊢Ⅎx∀y∈B¬yRz
8 5 7 nfan ⊢Ⅎxz∈B∧∀y∈B¬yRz
9 eleq1w ⊢x=z→x∈B↔z∈B
10 breq2 ⊢x=z→yRx↔yRz
11 10 notbid ⊢x=z→¬yRx↔¬yRz
12 11 ralbidv ⊢x=z→∀y∈B¬yRx↔∀y∈B¬yRz
13 9 12 anbi12d ⊢x=z→x∈B∧∀y∈B¬yRx↔z∈B∧∀y∈B¬yRz
14 3 8 13 cbvexv1 ⊢∃xx∈B∧∀y∈B¬yRx↔∃zz∈B∧∀y∈B¬yRz
15 df-rex ⊢∃x∈B∀y∈B¬yRx↔∃xx∈B∧∀y∈B¬yRx
16 df-rex ⊢∃z∈B∀y∈B¬yRz↔∃zz∈B∧∀y∈B¬yRz
17 14 15 16 3bitr4i ⊢∃x∈B∀y∈B¬yRx↔∃z∈B∀y∈B¬yRz
18 2 17 sylibr ⊢RFrSeA∧B⊆A∧B≠∅→∃x∈B∀y∈B¬yRx