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Database ZF (ZERMELO-FRAENKEL) SET THEORY ZF Set Theory - start with the Axiom of Extensionality Restricted quantification Restricted universal and existential quantification r19.45v
Description: Restricted quantifier version of one direction of 19.45 . The other
direction holds when A is nonempty, see r19.45zv . (Contributed by NM , 2-Apr-2004)
Ref
Expression
Assertion
r19.45v ⊢ ∃ x ∈ A φ ∨ ψ → φ ∨ ∃ x ∈ A ψ
Proof
Step
Hyp
Ref
Expression
1
r19.43 ⊢ ∃ x ∈ A φ ∨ ψ ↔ ∃ x ∈ A φ ∨ ∃ x ∈ A ψ
2
id ⊢ φ → φ
3
2
rexlimivw ⊢ ∃ x ∈ A φ → φ
4
3
orim1i ⊢ ∃ x ∈ A φ ∨ ∃ x ∈ A ψ → φ ∨ ∃ x ∈ A ψ
5
1 4
sylbi ⊢ ∃ x ∈ A φ ∨ ψ → φ ∨ ∃ x ∈ A ψ