Metamath Proof Explorer

Theorem bnj93

Description: Technical lemma for bnj97 . This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

Ref Expression
Assertion bnj93 ⊢ R FrSe A ∧ x ∈ A → pred x A R ∈ V

Proof

Step Hyp Ref Expression
1 df-bnj15 ⊢ R FrSe A ↔ R Fr A ∧ R Se A
2 1 simprbi ⊢ R FrSe A → R Se A
3 df-bnj13 ⊢ R Se A ↔ ∀ x ∈ A pred x A R ∈ V
4 2 3 sylib ⊢ R FrSe A → ∀ x ∈ A pred x A R ∈ V
5 4 r19.21bi ⊢ R FrSe A ∧ x ∈ A → pred x A R ∈ V