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Metamath Proof Explorer

Theorem tz6.26i

Description: All nonempty subclasses of a class having a well-ordered set-like relation R have R-minimal elements. Proposition 6.26 of TakeutiZaring p. 31. (Contributed by Scott Fenton, 14-Apr-2011) (Revised by Mario Carneiro, 26-Jun-2015)

	Ref	Expression
Hypotheses	tz6.26i.1	⊢ 𝑅 We 𝐴
	tz6.26i.2	⊢ 𝑅 Se 𝐴
Assertion	tz6.26i	⊢ ( ( 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅ ) → ∃ 𝑦 ∈ 𝐵 Pred ( 𝑅 , 𝐵 , 𝑦 ) = ∅ )

Proof

Step	Hyp	Ref	Expression
1		tz6.26i.1	⊢ 𝑅 We 𝐴
2		tz6.26i.2	⊢ 𝑅 Se 𝐴
3		tz6.26	⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅ ) ) → ∃ 𝑦 ∈ 𝐵 Pred ( 𝑅 , 𝐵 , 𝑦 ) = ∅ )
4	1 2 3	mpanl12	⊢ ( ( 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅ ) → ∃ 𝑦 ∈ 𝐵 Pred ( 𝑅 , 𝐵 , 𝑦 ) = ∅ )