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 )