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Metamath Proof Explorer

Theorem tz6.26

Description: All nonempty subclasses of a class having a well-ordered set-like relation have minimal elements for that relation. Proposition 6.26 of TakeutiZaring p. 31. (Contributed by Scott Fenton, 29-Jan-2011) (Revised by Mario Carneiro, 26-Jun-2015) (Proof shortened by Scott Fenton, 17-Nov-2024)

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

Proof

Step Hyp Ref Expression
1 wefr ⊢ ( 𝑅 We 𝐴 → 𝑅 Fr 𝐴 )
2 1 adantr ⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → 𝑅 Fr 𝐴 )
3 weso ⊢ ( 𝑅 We 𝐴 → 𝑅 Or 𝐴 )
4 sopo ⊢ ( 𝑅 Or 𝐴 → 𝑅 Po 𝐴 )
5 3 4 syl ⊢ ( 𝑅 We 𝐴 → 𝑅 Po 𝐴 )
6 5 adantr ⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → 𝑅 Po 𝐴 )
7 simpr ⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → 𝑅 Se 𝐴 )
8 2 6 7 3jca ⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) )
9 frpomin2 ⊢ ( ( ( 𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅ ) ) → ∃ 𝑦 ∈ 𝐵 Pred ( 𝑅 , 𝐵 , 𝑦 ) = ∅ )
10 8 9 sylan ⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅ ) ) → ∃ 𝑦 ∈ 𝐵 Pred ( 𝑅 , 𝐵 , 𝑦 ) = ∅ )