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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 → ∀ 𝑥 𝑤 ∈ 𝐵 )
Assertion bnj1228 ( ( 𝑅 FrSe 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅ ) → ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝑥 )

Proof

Step Hyp Ref Expression
1 bnj1228.1 ⊢ ( 𝑤 ∈ 𝐵 → ∀ 𝑥 𝑤 ∈ 𝐵 )
2 bnj69 ⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅ ) → ∃ 𝑧 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝑧 )
3 nfv ⊢ Ⅎ 𝑧 ( 𝑥 ∈ 𝐵 ∧ ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝑥 )
4 1 nfcii ⊢ Ⅎ 𝑥 𝐵
5 4 nfcri ⊢ Ⅎ 𝑥 𝑧 ∈ 𝐵
6 nfv ⊢ Ⅎ 𝑥 ¬ 𝑦 𝑅 𝑧
7 4 6 nfralw ⊢ Ⅎ 𝑥 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝑧
8 5 7 nfan ⊢ Ⅎ 𝑥 ( 𝑧 ∈ 𝐵 ∧ ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝑧 )
9 eleq1w ⊢ ( 𝑥 = 𝑧 → ( 𝑥 ∈ 𝐵 ↔ 𝑧 ∈ 𝐵 ) )
10 breq2 ⊢ ( 𝑥 = 𝑧 → ( 𝑦 𝑅 𝑥 ↔ 𝑦 𝑅 𝑧 ) )
11 10 notbid ⊢ ( 𝑥 = 𝑧 → ( ¬ 𝑦 𝑅 𝑥 ↔ ¬ 𝑦 𝑅 𝑧 ) )
12 11 ralbidv ⊢ ( 𝑥 = 𝑧 → ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝑥 ↔ ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝑧 ) )
13 9 12 anbi12d ⊢ ( 𝑥 = 𝑧 → ( ( 𝑥 ∈ 𝐵 ∧ ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝑥 ) ↔ ( 𝑧 ∈ 𝐵 ∧ ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝑧 ) ) )
14 3 8 13 cbvexv1 ⊢ ( ∃ 𝑥 ( 𝑥 ∈ 𝐵 ∧ ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝑥 ) ↔ ∃ 𝑧 ( 𝑧 ∈ 𝐵 ∧ ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝑧 ) )
15 df-rex ⊢ ( ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝑥 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐵 ∧ ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝑥 ) )
16 df-rex ⊢ ( ∃ 𝑧 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝑧 ↔ ∃ 𝑧 ( 𝑧 ∈ 𝐵 ∧ ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝑧 ) )
17 14 15 16 3bitr4i ⊢ ( ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝑥 ↔ ∃ 𝑧 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝑧 )
18 2 17 sylibr ⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅ ) → ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝑥 )